Main:z from

Percentage Accurate: 91.8% → 99.3%

Time: 26.0s

Alternatives: 22

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \sqrt{y + 1}\\ t_6 := t\_4 + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 1.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + t\_2}\right) + t\_4\right) + \left(t\_1 - \sqrt{t}\right)\\ \mathbf{elif}\;t\_6 \leq 2.1:\\ \;\;\;\;\left(t\_2 + \left(t\_5 + \frac{1}{\sqrt{z} + t\_3}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_2\right) + \left(t\_3 - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 t)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ t_4 (+ (- t_5 (sqrt y)) (- t_2 (sqrt x))))))
   (if (<= t_6 1.00005)
     (+
      (+ (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 (+ (sqrt x) t_2))) t_4)
      (- t_1 (sqrt t)))
     (if (<= t_6 2.1)
       (- (+ t_2 (+ t_5 (/ 1.0 (+ (sqrt z) t_3)))) (+ (sqrt y) (sqrt x)))
       (+
        (+ (+ 1.0 t_2) (- t_3 (+ (sqrt x) (+ (sqrt y) (sqrt z)))))
        (/ (- (+ 1.0 t) t) (+ (sqrt t) t_1)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + t));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = sqrt((y + 1.0));
	double t_6 = t_4 + ((t_5 - sqrt(y)) + (t_2 - sqrt(x)));
	double tmp;
	if (t_6 <= 1.00005) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (1.0 / (sqrt(x) + t_2))) + t_4) + (t_1 - sqrt(t));
	} else if (t_6 <= 2.1) {
		tmp = (t_2 + (t_5 + (1.0 / (sqrt(z) + t_3)))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((1.0 + t_2) + (t_3 - (sqrt(x) + (sqrt(y) + sqrt(z))))) + (((1.0 + t) - t) / (sqrt(t) + t_1));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + t))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(t_4 + Float64(Float64(t_5 - sqrt(y)) + Float64(t_2 - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 1.00005)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / Float64(sqrt(x) + t_2))) + t_4) + Float64(t_1 - sqrt(t)));
	elseif (t_6 <= 2.1)
		tmp = Float64(Float64(t_2 + Float64(t_5 + Float64(1.0 / Float64(sqrt(z) + t_3)))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 + t_2) + Float64(t_3 - Float64(sqrt(x) + Float64(sqrt(y) + sqrt(z))))) + Float64(Float64(Float64(1.0 + t) - t) / Float64(sqrt(t) + t_1)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.00005], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.1], N[(N[(t$95$2 + N[(t$95$5 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + t$95$2), $MachinePrecision] + N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 89.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 89.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 89.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 89.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 91.7%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x} + \sqrt{1 + x}\right) \cdot 1\right)}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 71.4

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 71.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.10000000000000009

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 95.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 95.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 95.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Simplified 26.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2.10000000000000009 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 95.6

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{\color{blue}{t + 1}} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\color{blue}{\sqrt{t + 1}} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \color{blue}{\sqrt{t}}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\left(1 + t\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\color{blue}{\left(1 + t\right)} - t}{\sqrt{t + 1} + \sqrt{t}} \]

      16. lower-+.f64 95.6

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\color{blue}{\sqrt{t + 1} + \sqrt{t}}} \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{t + 1}} + \sqrt{t}} \]

      18. +-commutative N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{1 + t}} + \sqrt{t}} \]

      19. lower-+.f64 95.6

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{\color{blue}{1 + t}} + \sqrt{t}} \]

   7. Applied egg-rr 95.6%

      \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \color{blue}{\frac{\left(1 + t\right) - t}{\sqrt{1 + t} + \sqrt{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.1:\\ \;\;\;\;\left(\sqrt{1 + x} + \left(\sqrt{y + 1} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \frac{\left(1 + t\right) - t}{\sqrt{t} + \sqrt{1 + t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{y} + \sqrt{x}\\ t_3 := \sqrt{1 + x}\\ t_4 := \sqrt{1 + z}\\ t_5 := \sqrt{y + 1}\\ t_6 := \left(t\_4 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + t\_1\right)\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + t\_3}, \frac{0.5}{\sqrt{z}} + \frac{0.5}{\sqrt{y}}\right)\\ \mathbf{elif}\;t\_6 \leq 2.00005:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, t\_1, t\_5\right) - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(t\_4 - \left(\sqrt{z} + t\_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 z)))
        (t_2 (+ (sqrt y) (sqrt x)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (sqrt (+ 1.0 z)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ (- t_4 (sqrt z)) (+ (- t_5 (sqrt y)) (- t_3 (sqrt x))))))
   (if (<= t_6 0.0)
     (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) t_1)))
     (if (<= t_6 1.00005)
       (fma
        (- (+ 1.0 x) x)
        (/ 1.0 (+ (sqrt x) t_3))
        (+ (/ 0.5 (sqrt z)) (/ 0.5 (sqrt y))))
       (if (<= t_6 2.00005)
         (+ t_3 (- (fma 0.5 t_1 t_5) t_2))
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ 2.0 (- t_4 (+ (sqrt z) t_2)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double t_2 = sqrt(y) + sqrt(x);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + z));
	double t_5 = sqrt((y + 1.0));
	double t_6 = (t_4 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_3 - sqrt(x)));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = 0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + t_1));
	} else if (t_6 <= 1.00005) {
		tmp = fma(((1.0 + x) - x), (1.0 / (sqrt(x) + t_3)), ((0.5 / sqrt(z)) + (0.5 / sqrt(y))));
	} else if (t_6 <= 2.00005) {
		tmp = t_3 + (fma(0.5, t_1, t_5) - t_2);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (2.0 + (t_4 - (sqrt(z) + t_2)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	t_2 = Float64(sqrt(y) + sqrt(x))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = sqrt(Float64(1.0 + z))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_5 - sqrt(y)) + Float64(t_3 - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + t_1)));
	elseif (t_6 <= 1.00005)
		tmp = fma(Float64(Float64(1.0 + x) - x), Float64(1.0 / Float64(sqrt(x) + t_3)), Float64(Float64(0.5 / sqrt(z)) + Float64(0.5 / sqrt(y))));
	elseif (t_6 <= 2.00005)
		tmp = Float64(t_3 + Float64(fma(0.5, t_1, t_5) - t_2));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(2.0 + Float64(t_4 - Float64(sqrt(z) + t_2))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.00005], N[(t$95$3 + N[(N[(0.5 * t$95$1 + t$95$5), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(t$95$4 - N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 71.8

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 71.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 58.6

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 58.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.2

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 28.2

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 28.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 44.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   13. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}} + \frac{0.5}{\sqrt{y}}\right)} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0000499999999999

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      15. lower-sqrt.f64 22.8

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 22.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.0000499999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 94.5

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 87.3

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(\left(1 + x\right) - x, \frac{1}{\sqrt{x} + \sqrt{1 + x}}, \frac{0.5}{\sqrt{z}} + \frac{0.5}{\sqrt{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.00005:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_3 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\\ t_6 := \sqrt{\frac{1}{y}} + t\_1\\ t_7 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_6\right)\\ \mathbf{elif}\;t\_5 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_6, t\_2\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, t\_1, t\_4\right) - t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(t\_3 - \left(\sqrt{z} + t\_7\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_3 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_2 (sqrt x)))))
        (t_6 (+ (sqrt (/ 1.0 y)) t_1))
        (t_7 (+ (sqrt y) (sqrt x))))
   (if (<= t_5 5e-5)
     (* 0.5 (+ (sqrt (/ 1.0 x)) t_6))
     (if (<= t_5 1.00005)
       (- (fma 0.5 t_6 t_2) (sqrt x))
       (if (<= t_5 2.00005)
         (+ t_2 (- (fma 0.5 t_1 t_4) t_7))
         (+
          (- (sqrt (+ 1.0 t)) (sqrt t))
          (+ 2.0 (- t_3 (+ (sqrt z) t_7)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_3 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_2 - sqrt(x)));
	double t_6 = sqrt((1.0 / y)) + t_1;
	double t_7 = sqrt(y) + sqrt(x);
	double tmp;
	if (t_5 <= 5e-5) {
		tmp = 0.5 * (sqrt((1.0 / x)) + t_6);
	} else if (t_5 <= 1.00005) {
		tmp = fma(0.5, t_6, t_2) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = t_2 + (fma(0.5, t_1, t_4) - t_7);
	} else {
		tmp = (sqrt((1.0 + t)) - sqrt(t)) + (2.0 + (t_3 - (sqrt(z) + t_7)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_2 - sqrt(x))))
	t_6 = Float64(sqrt(Float64(1.0 / y)) + t_1)
	t_7 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (t_5 <= 5e-5)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_6));
	elseif (t_5 <= 1.00005)
		tmp = Float64(fma(0.5, t_6, t_2) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(t_2 + Float64(fma(0.5, t_1, t_4) - t_7));
	else
		tmp = Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) + Float64(2.0 + Float64(t_3 - Float64(sqrt(z) + t_7))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-5], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.00005], N[(N[(0.5 * t$95$6 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(t$95$2 + N[(N[(0.5 * t$95$1 + t$95$4), $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(t$95$3 - N[(N[Sqrt[z], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.00000000000000024e-5

   1. Initial program 60.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 72.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 72.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 59.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 26.9

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 5.00000000000000024e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 67.9

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 67.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 43.7

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.7

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0000499999999999

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      15. lower-sqrt.f64 22.8

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 22.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.0000499999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 94.5

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 87.3

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.00005:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{z}}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_3 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right)\\ t_6 := \sqrt{\frac{1}{y}} + t\_1\\ t_7 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_6\right)\\ \mathbf{elif}\;t\_5 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_6, t\_2\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, t\_1, t\_4\right) - t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_3 + t\_4\right)\right) - \left(\sqrt{z} + t\_7\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_3 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_2 (sqrt x)))))
        (t_6 (+ (sqrt (/ 1.0 y)) t_1))
        (t_7 (+ (sqrt y) (sqrt x))))
   (if (<= t_5 5e-5)
     (* 0.5 (+ (sqrt (/ 1.0 x)) t_6))
     (if (<= t_5 1.00005)
       (- (fma 0.5 t_6 t_2) (sqrt x))
       (if (<= t_5 2.00005)
         (+ t_2 (- (fma 0.5 t_1 t_4) t_7))
         (- (+ 1.0 (+ t_3 t_4)) (+ (sqrt z) t_7)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 / z));
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_3 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_2 - sqrt(x)));
	double t_6 = sqrt((1.0 / y)) + t_1;
	double t_7 = sqrt(y) + sqrt(x);
	double tmp;
	if (t_5 <= 5e-5) {
		tmp = 0.5 * (sqrt((1.0 / x)) + t_6);
	} else if (t_5 <= 1.00005) {
		tmp = fma(0.5, t_6, t_2) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = t_2 + (fma(0.5, t_1, t_4) - t_7);
	} else {
		tmp = (1.0 + (t_3 + t_4)) - (sqrt(z) + t_7);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 / z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_2 - sqrt(x))))
	t_6 = Float64(sqrt(Float64(1.0 / y)) + t_1)
	t_7 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (t_5 <= 5e-5)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_6));
	elseif (t_5 <= 1.00005)
		tmp = Float64(fma(0.5, t_6, t_2) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(t_2 + Float64(fma(0.5, t_1, t_4) - t_7));
	else
		tmp = Float64(Float64(1.0 + Float64(t_3 + t_4)) - Float64(sqrt(z) + t_7));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-5], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.00005], N[(N[(0.5 * t$95$6 + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(t$95$2 + N[(N[(0.5 * t$95$1 + t$95$4), $MachinePrecision] - t$95$7), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.00000000000000024e-5

   1. Initial program 60.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 72.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 72.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 59.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 26.9

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 5.00000000000000024e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 67.9

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 67.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 43.7

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.7

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0000499999999999

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      15. lower-sqrt.f64 22.8

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 22.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.0000499999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 59.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right) \]

      15. lower-sqrt.f64 51.2

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 51.2%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.00005:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \sqrt{y + 1}\right)\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{y} + \sqrt{x}\\ t_4 := \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\\ t_5 := \sqrt{y + 1}\\ t_6 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + t\_4\right)\\ \mathbf{elif}\;t\_6 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_4, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_5 - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_2 + t\_5\right)\right) - \left(\sqrt{z} + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (+ (sqrt y) (sqrt x)))
        (t_4 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ (- t_2 (sqrt z)) (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_6 5e-5)
     (* 0.5 (+ (sqrt (/ 1.0 x)) t_4))
     (if (<= t_6 1.00005)
       (- (fma 0.5 t_4 t_1) (sqrt x))
       (if (<= t_6 2.0)
         (+ t_1 (- t_5 t_3))
         (- (+ 1.0 (+ t_2 t_5)) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt(y) + sqrt(x);
	double t_4 = sqrt((1.0 / y)) + sqrt((1.0 / z));
	double t_5 = sqrt((y + 1.0));
	double t_6 = (t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_6 <= 5e-5) {
		tmp = 0.5 * (sqrt((1.0 / x)) + t_4);
	} else if (t_6 <= 1.00005) {
		tmp = fma(0.5, t_4, t_1) - sqrt(x);
	} else if (t_6 <= 2.0) {
		tmp = t_1 + (t_5 - t_3);
	} else {
		tmp = (1.0 + (t_2 + t_5)) - (sqrt(z) + t_3);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(y) + sqrt(x))
	t_4 = Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z)))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 5e-5)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + t_4));
	elseif (t_6 <= 1.00005)
		tmp = Float64(fma(0.5, t_4, t_1) - sqrt(x));
	elseif (t_6 <= 2.0)
		tmp = Float64(t_1 + Float64(t_5 - t_3));
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 + t_5)) - Float64(sqrt(z) + t_3));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 5e-5], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.00005], N[(N[(0.5 * t$95$4 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(t$95$1 + N[(t$95$5 - t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.00000000000000024e-5

   1. Initial program 60.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 72.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 72.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 59.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 26.9

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 5.00000000000000024e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 67.9

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 67.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 43.7

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.7

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 95.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 25.5

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 58.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right) \]

      15. lower-sqrt.f64 50.3

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 50.3%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \sqrt{y + 1}\right)\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{y} + \sqrt{x}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ t_6 := \sqrt{\frac{1}{y}}\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(t\_6 + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_6, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_4 - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_2 + t\_4\right)\right) - \left(\sqrt{z} + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (+ (sqrt y) (sqrt x)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_1 (sqrt x)))))
        (t_6 (sqrt (/ 1.0 y))))
   (if (<= t_5 5e-5)
     (* 0.5 (+ (sqrt (/ 1.0 x)) (+ t_6 (sqrt (/ 1.0 z)))))
     (if (<= t_5 1.00005)
       (- (fma 0.5 t_6 t_1) (sqrt x))
       (if (<= t_5 2.0)
         (+ t_1 (- t_4 t_3))
         (- (+ 1.0 (+ t_2 t_4)) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt(y) + sqrt(x);
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double t_6 = sqrt((1.0 / y));
	double tmp;
	if (t_5 <= 5e-5) {
		tmp = 0.5 * (sqrt((1.0 / x)) + (t_6 + sqrt((1.0 / z))));
	} else if (t_5 <= 1.00005) {
		tmp = fma(0.5, t_6, t_1) - sqrt(x);
	} else if (t_5 <= 2.0) {
		tmp = t_1 + (t_4 - t_3);
	} else {
		tmp = (1.0 + (t_2 + t_4)) - (sqrt(z) + t_3);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(y) + sqrt(x))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	t_6 = sqrt(Float64(1.0 / y))
	tmp = 0.0
	if (t_5 <= 5e-5)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(t_6 + sqrt(Float64(1.0 / z)))));
	elseif (t_5 <= 1.00005)
		tmp = Float64(fma(0.5, t_6, t_1) - sqrt(x));
	elseif (t_5 <= 2.0)
		tmp = Float64(t_1 + Float64(t_4 - t_3));
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 + t_4)) - Float64(sqrt(z) + t_3));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 5e-5], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(t$95$6 + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.00005], N[(N[(0.5 * t$95$6 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(t$95$1 + N[(t$95$4 - t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.00000000000000024e-5

   1. Initial program 60.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 72.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 72.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 59.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 26.9

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 5.00000000000000024e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 67.9

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 67.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 43.7

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 43.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.7

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.9

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.9%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 95.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 25.5

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 58.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right) \]

      15. lower-sqrt.f64 50.3

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 50.3%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \sqrt{y + 1}\right)\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{y} + \sqrt{x}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_5 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_4 - t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(t\_2 + t\_4\right)\right) - \left(\sqrt{z} + t\_3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (+ (sqrt y) (sqrt x)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_5 0.0)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_5 1.00005)
       (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
       (if (<= t_5 2.0)
         (+ t_1 (- t_4 t_3))
         (- (+ 1.0 (+ t_2 t_4)) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt(y) + sqrt(x);
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_5 <= 1.00005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else if (t_5 <= 2.0) {
		tmp = t_1 + (t_4 - t_3);
	} else {
		tmp = (1.0 + (t_2 + t_4)) - (sqrt(z) + t_3);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(y) + sqrt(x))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_5 <= 1.00005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	elseif (t_5 <= 2.0)
		tmp = Float64(t_1 + Float64(t_4 - t_3));
	else
		tmp = Float64(Float64(1.0 + Float64(t_2 + t_4)) - Float64(sqrt(z) + t_3));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.00005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(t$95$1 + N[(t$95$4 - t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 60.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 60.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 44.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.6

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 95.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 25.5

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 58.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right) \]

      15. lower-sqrt.f64 50.3

          \[\leadsto \left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right) \]

   8. Simplified 50.3%

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + z} + \sqrt{1 + y}\right)\right) - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + z} + \sqrt{y + 1}\right)\right) - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{y + 1}\\ t_4 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_3 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + t\_3\right) + \left(t\_1 - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_4 0.0)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_4 1.00005)
       (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
       (if (<= t_4 2.0)
         (+ t_1 (- t_3 (+ (sqrt y) (sqrt x))))
         (+ (+ t_2 t_3) (- t_1 (+ (sqrt x) (sqrt z)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((y + 1.0));
	double t_4 = (t_2 - sqrt(z)) + ((t_3 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_4 <= 1.00005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = t_1 + (t_3 - (sqrt(y) + sqrt(x)));
	} else {
		tmp = (t_2 + t_3) + (t_1 - (sqrt(x) + sqrt(z)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(t_1 + Float64(t_3 - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(Float64(t_2 + t_3) + Float64(t_1 - Float64(sqrt(x) + sqrt(z))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(t$95$1 + N[(t$95$3 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + t$95$3), $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 60.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 60.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 44.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.6

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 95.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 27.2

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 25.5

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 25.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 58.4

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 54.3

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 54.3%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + z} + \sqrt{y + 1}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{y} + \sqrt{x}\\ t_3 := \sqrt{y + 1}\\ t_4 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_4 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.5:\\ \;\;\;\;t\_1 + \left(t\_3 - t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_3\right) + \left(t\_1 - t\_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (+ (sqrt y) (sqrt x)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4
         (+
          (- (sqrt (+ 1.0 z)) (sqrt z))
          (+ (- t_3 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_4 0.0)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_4 1.00005)
       (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
       (if (<= t_4 2.5) (+ t_1 (- t_3 t_2)) (+ (+ 1.0 t_3) (- t_1 t_2)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt(y) + sqrt(x);
	double t_3 = sqrt((y + 1.0));
	double t_4 = (sqrt((1.0 + z)) - sqrt(z)) + ((t_3 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_4 <= 1.00005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else if (t_4 <= 2.5) {
		tmp = t_1 + (t_3 - t_2);
	} else {
		tmp = (1.0 + t_3) + (t_1 - t_2);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(sqrt(y) + sqrt(x))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(t_3 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_4 <= 1.00005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	elseif (t_4 <= 2.5)
		tmp = Float64(t_1 + Float64(t_3 - t_2));
	else
		tmp = Float64(Float64(1.0 + t_3) + Float64(t_1 - t_2));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.00005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.5], N[(t$95$1 + N[(t$95$3 - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$3), $MachinePrecision] + N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 60.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 60.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 44.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.6

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.5

   1. Initial program 95.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 29.3

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 29.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 25.2

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 25.2%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 56.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 52.0

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 52.0%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \color{blue}{\sqrt{1 + y}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       3. lower-+.f64 52.0

          \[\leadsto \left(1 + \sqrt{\color{blue}{1 + y}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   11. Simplified 52.0%

       \[\leadsto \color{blue}{\left(1 + \sqrt{1 + y}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.5:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{y + 1}\right) + \left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), -0.125 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\\ \mathbf{elif}\;t\_5 \leq 1.99999995:\\ \;\;\;\;t\_3 + \left(\left(t\_1 + \frac{1}{\sqrt{y} + t\_4}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(1 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_5 0.0002)
     (fma
      0.5
      (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))))
      (* -0.125 (sqrt (/ 1.0 (* x (* x x))))))
     (if (<= t_5 1.99999995)
       (+ t_3 (- (+ t_1 (/ 1.0 (+ (sqrt y) t_4))) (sqrt x)))
       (+
        t_3
        (+ 1.0 (- (+ t_1 (/ 1.0 (+ (sqrt z) t_2))) (+ (sqrt y) (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_5 <= 0.0002) {
		tmp = fma(0.5, (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z)))), (-0.125 * sqrt((1.0 / (x * (x * x))))));
	} else if (t_5 <= 1.99999995) {
		tmp = t_3 + ((t_1 + (1.0 / (sqrt(y) + t_4))) - sqrt(x));
	} else {
		tmp = t_3 + (1.0 + ((t_1 + (1.0 / (sqrt(z) + t_2))) - (sqrt(y) + sqrt(x))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_5 <= 0.0002)
		tmp = fma(0.5, Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z)))), Float64(-0.125 * sqrt(Float64(1.0 / Float64(x * Float64(x * x))))));
	elseif (t_5 <= 1.99999995)
		tmp = Float64(t_3 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(y) + t_4))) - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(1.0 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(y) + sqrt(x)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0002], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.99999995], N[(t$95$3 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(1.0 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0000000000000001e-4

   1. Initial program 60.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 71.8

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 71.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 56.9

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) + \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

       2. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} + \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}, \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}, \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       9. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       10. lower-sqrt.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       11. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}}\right) \]

       13. lower-sqrt.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}\right) \]

       14. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}\right) \]

       15. cube-mult N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), \frac{-1}{8} \cdot \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}\right) \]

       17. lower-*.f64 28.1

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), -0.125 \cdot \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}\right) \]

   14. Simplified 28.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), -0.125 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)} \]

   if 2.0000000000000001e-4 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.99999995000000008

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 46.7

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.99999995000000008 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(1 + \left(\left(\color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 56.5

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right), -0.125 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.99999995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_3 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_5 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 1.99999995:\\ \;\;\;\;t\_2 + \left(\left(t\_1 + \frac{1}{\sqrt{y} + t\_4}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(1 + \left(\left(t\_1 + \frac{1}{\sqrt{z} + t\_3}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_3 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_5 5e-5)
     (* 0.5 (+ (sqrt (/ 1.0 x)) (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z)))))
     (if (<= t_5 1.99999995)
       (+ t_2 (- (+ t_1 (/ 1.0 (+ (sqrt y) t_4))) (sqrt x)))
       (+
        t_2
        (+ 1.0 (- (+ t_1 (/ 1.0 (+ (sqrt z) t_3))) (+ (sqrt y) (sqrt x)))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_3 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_5 <= 5e-5) {
		tmp = 0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z))));
	} else if (t_5 <= 1.99999995) {
		tmp = t_2 + ((t_1 + (1.0 / (sqrt(y) + t_4))) - sqrt(x));
	} else {
		tmp = t_2 + (1.0 + ((t_1 + (1.0 / (sqrt(z) + t_3))) - (sqrt(y) + sqrt(x))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + t)) - sqrt(t)
    t_3 = sqrt((1.0d0 + z))
    t_4 = sqrt((y + 1.0d0))
    t_5 = (t_3 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)))
    if (t_5 <= 5d-5) then
        tmp = 0.5d0 * (sqrt((1.0d0 / x)) + (sqrt((1.0d0 / y)) + sqrt((1.0d0 / z))))
    else if (t_5 <= 1.99999995d0) then
        tmp = t_2 + ((t_1 + (1.0d0 / (sqrt(y) + t_4))) - sqrt(x))
    else
        tmp = t_2 + (1.0d0 + ((t_1 + (1.0d0 / (sqrt(z) + t_3))) - (sqrt(y) + sqrt(x))))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_3 = Math.sqrt((1.0 + z));
	double t_4 = Math.sqrt((y + 1.0));
	double t_5 = (t_3 - Math.sqrt(z)) + ((t_4 - Math.sqrt(y)) + (t_1 - Math.sqrt(x)));
	double tmp;
	if (t_5 <= 5e-5) {
		tmp = 0.5 * (Math.sqrt((1.0 / x)) + (Math.sqrt((1.0 / y)) + Math.sqrt((1.0 / z))));
	} else if (t_5 <= 1.99999995) {
		tmp = t_2 + ((t_1 + (1.0 / (Math.sqrt(y) + t_4))) - Math.sqrt(x));
	} else {
		tmp = t_2 + (1.0 + ((t_1 + (1.0 / (Math.sqrt(z) + t_3))) - (Math.sqrt(y) + Math.sqrt(x))));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_3 = math.sqrt((1.0 + z))
	t_4 = math.sqrt((y + 1.0))
	t_5 = (t_3 - math.sqrt(z)) + ((t_4 - math.sqrt(y)) + (t_1 - math.sqrt(x)))
	tmp = 0
	if t_5 <= 5e-5:
		tmp = 0.5 * (math.sqrt((1.0 / x)) + (math.sqrt((1.0 / y)) + math.sqrt((1.0 / z))))
	elif t_5 <= 1.99999995:
		tmp = t_2 + ((t_1 + (1.0 / (math.sqrt(y) + t_4))) - math.sqrt(x))
	else:
		tmp = t_2 + (1.0 + ((t_1 + (1.0 / (math.sqrt(z) + t_3))) - (math.sqrt(y) + math.sqrt(x))))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_5 <= 5e-5)
		tmp = Float64(0.5 * Float64(sqrt(Float64(1.0 / x)) + Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z)))));
	elseif (t_5 <= 1.99999995)
		tmp = Float64(t_2 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(y) + t_4))) - sqrt(x)));
	else
		tmp = Float64(t_2 + Float64(1.0 + Float64(Float64(t_1 + Float64(1.0 / Float64(sqrt(z) + t_3))) - Float64(sqrt(y) + sqrt(x)))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + t)) - sqrt(t);
	t_3 = sqrt((1.0 + z));
	t_4 = sqrt((y + 1.0));
	t_5 = (t_3 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	tmp = 0.0;
	if (t_5 <= 5e-5)
		tmp = 0.5 * (sqrt((1.0 / x)) + (sqrt((1.0 / y)) + sqrt((1.0 / z))));
	elseif (t_5 <= 1.99999995)
		tmp = t_2 + ((t_1 + (1.0 / (sqrt(y) + t_4))) - sqrt(x));
	else
		tmp = t_2 + (1.0 + ((t_1 + (1.0 / (sqrt(z) + t_3))) - (sqrt(y) + sqrt(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 5e-5], N[(0.5 * N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.99999995], N[(t$95$2 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(1.0 + N[(N[(t$95$1 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 5.00000000000000024e-5

   1. Initial program 60.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 72.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 72.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 59.1

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 59.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 3.6

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 3.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)} \]
   13. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\sqrt{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\color{blue}{\frac{1}{x}}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}\right)\right) \]

       10. lower-/.f64 26.9

           \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}\right)\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)} \]

   if 5.00000000000000024e-5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.99999995000000008

   1. Initial program 96.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.6

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.6

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.6%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 46.3

          \[\leadsto \left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 46.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.99999995000000008 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(1 + \left(\left(\color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 56.5

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 56.5%

      \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{1}{x}} + \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.99999995:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{y + 1}\\ t_5 := t\_3 + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ t_6 := \sqrt{1 + t} - \sqrt{t}\\ t_7 := \sqrt{y} + \sqrt{x}\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;t\_6 + \left(\frac{1}{\sqrt{x} + t\_1} + t\_3\right)\\ \mathbf{elif}\;t\_5 \leq 2.1:\\ \;\;\;\;\left(t\_1 + \left(t\_4 + \frac{1}{\sqrt{z} + t\_2}\right)\right) - t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6 + \left(2 + \left(t\_2 - \left(\sqrt{z} + t\_7\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ t_3 (+ (- t_4 (sqrt y)) (- t_1 (sqrt x)))))
        (t_6 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_7 (+ (sqrt y) (sqrt x))))
   (if (<= t_5 1.0)
     (+ t_6 (+ (/ 1.0 (+ (sqrt x) t_1)) t_3))
     (if (<= t_5 2.1)
       (- (+ t_1 (+ t_4 (/ 1.0 (+ (sqrt z) t_2)))) t_7)
       (+ t_6 (+ 2.0 (- t_2 (+ (sqrt z) t_7))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((y + 1.0));
	double t_5 = t_3 + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double t_6 = sqrt((1.0 + t)) - sqrt(t);
	double t_7 = sqrt(y) + sqrt(x);
	double tmp;
	if (t_5 <= 1.0) {
		tmp = t_6 + ((1.0 / (sqrt(x) + t_1)) + t_3);
	} else if (t_5 <= 2.1) {
		tmp = (t_1 + (t_4 + (1.0 / (sqrt(z) + t_2)))) - t_7;
	} else {
		tmp = t_6 + (2.0 + (t_2 - (sqrt(z) + t_7)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + x))
    t_2 = sqrt((1.0d0 + z))
    t_3 = t_2 - sqrt(z)
    t_4 = sqrt((y + 1.0d0))
    t_5 = t_3 + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)))
    t_6 = sqrt((1.0d0 + t)) - sqrt(t)
    t_7 = sqrt(y) + sqrt(x)
    if (t_5 <= 1.0d0) then
        tmp = t_6 + ((1.0d0 / (sqrt(x) + t_1)) + t_3)
    else if (t_5 <= 2.1d0) then
        tmp = (t_1 + (t_4 + (1.0d0 / (sqrt(z) + t_2)))) - t_7
    else
        tmp = t_6 + (2.0d0 + (t_2 - (sqrt(z) + t_7)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((1.0 + x));
	double t_2 = Math.sqrt((1.0 + z));
	double t_3 = t_2 - Math.sqrt(z);
	double t_4 = Math.sqrt((y + 1.0));
	double t_5 = t_3 + ((t_4 - Math.sqrt(y)) + (t_1 - Math.sqrt(x)));
	double t_6 = Math.sqrt((1.0 + t)) - Math.sqrt(t);
	double t_7 = Math.sqrt(y) + Math.sqrt(x);
	double tmp;
	if (t_5 <= 1.0) {
		tmp = t_6 + ((1.0 / (Math.sqrt(x) + t_1)) + t_3);
	} else if (t_5 <= 2.1) {
		tmp = (t_1 + (t_4 + (1.0 / (Math.sqrt(z) + t_2)))) - t_7;
	} else {
		tmp = t_6 + (2.0 + (t_2 - (Math.sqrt(z) + t_7)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + x))
	t_2 = math.sqrt((1.0 + z))
	t_3 = t_2 - math.sqrt(z)
	t_4 = math.sqrt((y + 1.0))
	t_5 = t_3 + ((t_4 - math.sqrt(y)) + (t_1 - math.sqrt(x)))
	t_6 = math.sqrt((1.0 + t)) - math.sqrt(t)
	t_7 = math.sqrt(y) + math.sqrt(x)
	tmp = 0
	if t_5 <= 1.0:
		tmp = t_6 + ((1.0 / (math.sqrt(x) + t_1)) + t_3)
	elif t_5 <= 2.1:
		tmp = (t_1 + (t_4 + (1.0 / (math.sqrt(z) + t_2)))) - t_7
	else:
		tmp = t_6 + (2.0 + (t_2 - (math.sqrt(z) + t_7)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(t_3 + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	t_6 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_7 = Float64(sqrt(y) + sqrt(x))
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(t_6 + Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + t_3));
	elseif (t_5 <= 2.1)
		tmp = Float64(Float64(t_1 + Float64(t_4 + Float64(1.0 / Float64(sqrt(z) + t_2)))) - t_7);
	else
		tmp = Float64(t_6 + Float64(2.0 + Float64(t_2 - Float64(sqrt(z) + t_7))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((1.0 + x));
	t_2 = sqrt((1.0 + z));
	t_3 = t_2 - sqrt(z);
	t_4 = sqrt((y + 1.0));
	t_5 = t_3 + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	t_6 = sqrt((1.0 + t)) - sqrt(t);
	t_7 = sqrt(y) + sqrt(x);
	tmp = 0.0;
	if (t_5 <= 1.0)
		tmp = t_6 + ((1.0 / (sqrt(x) + t_1)) + t_3);
	elseif (t_5 <= 2.1)
		tmp = (t_1 + (t_4 + (1.0 / (sqrt(z) + t_2)))) - t_7;
	else
		tmp = t_6 + (2.0 + (t_2 - (sqrt(z) + t_7)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(t$95$6 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.1], N[(N[(t$95$1 + N[(t$95$4 + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$7), $MachinePrecision], N[(t$95$6 + N[(2.0 + N[(t$95$2 - N[(N[Sqrt[z], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1

   1. Initial program 90.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 90.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 90.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 90.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 92.2%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x} + \sqrt{1 + x}\right) \cdot 1\right)}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 71.6

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 71.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.10000000000000009

   1. Initial program 94.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 95.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 95.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 95.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + y}} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Simplified 25.9%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2.10000000000000009 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 95.6

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 95.6

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.1:\\ \;\;\;\;\left(\sqrt{1 + x} + \left(\sqrt{y + 1} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{y} + \sqrt{x}\\ t_6 := \sqrt{y + 1}\\ t_7 := t\_3 + \left(\left(t\_6 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_7 \leq 0.999999999793685:\\ \;\;\;\;t\_4 + \left(\frac{1}{\sqrt{x} + t\_1} + t\_3\right)\\ \mathbf{elif}\;t\_7 \leq 2.00005:\\ \;\;\;\;t\_1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_6\right) - t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(2 + \left(t\_2 - \left(\sqrt{z} + t\_5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (+ (sqrt y) (sqrt x)))
        (t_6 (sqrt (+ y 1.0)))
        (t_7 (+ t_3 (+ (- t_6 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_7 0.999999999793685)
     (+ t_4 (+ (/ 1.0 (+ (sqrt x) t_1)) t_3))
     (if (<= t_7 2.00005)
       (+ t_1 (- (fma 0.5 (sqrt (/ 1.0 z)) t_6) t_5))
       (+ t_4 (+ 2.0 (- t_2 (+ (sqrt z) t_5))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((1.0 + z));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt(y) + sqrt(x);
	double t_6 = sqrt((y + 1.0));
	double t_7 = t_3 + ((t_6 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_7 <= 0.999999999793685) {
		tmp = t_4 + ((1.0 / (sqrt(x) + t_1)) + t_3);
	} else if (t_7 <= 2.00005) {
		tmp = t_1 + (fma(0.5, sqrt((1.0 / z)), t_6) - t_5);
	} else {
		tmp = t_4 + (2.0 + (t_2 - (sqrt(z) + t_5)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = Float64(sqrt(y) + sqrt(x))
	t_6 = sqrt(Float64(y + 1.0))
	t_7 = Float64(t_3 + Float64(Float64(t_6 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_7 <= 0.999999999793685)
		tmp = Float64(t_4 + Float64(Float64(1.0 / Float64(sqrt(x) + t_1)) + t_3));
	elseif (t_7 <= 2.00005)
		tmp = Float64(t_1 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_6) - t_5));
	else
		tmp = Float64(t_4 + Float64(2.0 + Float64(t_2 - Float64(sqrt(z) + t_5))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$3 + N[(N[(t$95$6 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.999999999793685], N[(t$95$4 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.00005], N[(t$95$1 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(2.0 + N[(t$95$2 - N[(N[Sqrt[z], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.99999999979368503

   1. Initial program 65.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 65.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 65.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 65.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 72.9%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x} + \sqrt{1 + x}\right) \cdot 1\right)}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 66.7

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 66.7%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.99999999979368503 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0000499999999999

   1. Initial program 96.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 15.5

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 15.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      15. lower-sqrt.f64 24.5

          \[\leadsto \sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 24.5%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]

   if 2.0000499999999999 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 99.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + x} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(1 + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 94.5

          \[\leadsto \left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 94.5%

      \[\leadsto \color{blue}{\left(\left(1 + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \color{blue}{\sqrt{x}}\right) + \sqrt{z}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 87.3

          \[\leadsto \left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \color{blue}{\sqrt{z}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + z} - \left(\left(\sqrt{y} + \sqrt{x}\right) + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0.999999999793685:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.00005:\\ \;\;\;\;\sqrt{1 + x} + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(2 + \left(\sqrt{1 + z} - \left(\sqrt{z} + \left(\sqrt{y} + \sqrt{x}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{y + 1}\\ t_3 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_3 \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(t\_2 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3
         (+
          (- (sqrt (+ 1.0 z)) (sqrt z))
          (+ (- t_2 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_3 0.0)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_3 1.00005)
       (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
       (+ t_1 (- t_2 (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((y + 1.0));
	double t_3 = (sqrt((1.0 + z)) - sqrt(z)) + ((t_2 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_3 <= 1.00005) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else {
		tmp = t_1 + (t_2 - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(t_2 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_3 <= 1.00005)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	else
		tmp = Float64(t_1 + Float64(t_2 - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.00005], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t$95$2 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 60.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 60.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.00005000000000011

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.1

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 44.0

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 44.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.4

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.6

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.00005000000000011 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 35.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 35.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) \]

      11. lower-sqrt.f64 23.8

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) \]

   8. Simplified 23.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.00005:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{y + 1} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;t\_2 \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y} + \left(t\_1 - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 1.0 x)))
        (t_2
         (+
          (- (sqrt (+ 1.0 z)) (sqrt z))
          (+ (- (sqrt (+ y 1.0)) (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_2 0.0)
     (* 0.5 (sqrt (/ 1.0 x)))
     (if (<= t_2 1.5)
       (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x))
       (+ (sqrt y) (- t_1 (+ (sqrt y) (sqrt x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = (sqrt((1.0 + z)) - sqrt(z)) + ((sqrt((y + 1.0)) - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else if (t_2 <= 1.5) {
		tmp = fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x);
	} else {
		tmp = sqrt(y) + (t_1 - (sqrt(y) + sqrt(x)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) + Float64(Float64(sqrt(Float64(y + 1.0)) - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	elseif (t_2 <= 1.5)
		tmp = Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x));
	else
		tmp = Float64(sqrt(y) + Float64(t_1 - Float64(sqrt(y) + sqrt(x))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.5], N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[y], $MachinePrecision] + N[(t$95$1 - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 58.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 60.6

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 60.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.5

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 68.4

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 68.4%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right)} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{z}}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 43.4

          \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \sqrt{x}} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \sqrt{x} \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right)} - \sqrt{x} \]

       4. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       6. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \color{blue}{\sqrt{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\color{blue}{\frac{1}{z}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x} \]

       10. lower-+.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x} \]

       11. lower-sqrt.f64 27.3

           \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}} \]

   11. Simplified 27.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{1 + x}\right) - \sqrt{x}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

       2. lower-/.f64 25.6

          \[\leadsto \mathsf{fma}\left(0.5, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   14. Simplified 25.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x} \]

   if 1.5 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 37.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 37.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 36.5

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 36.5%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 11.8

          \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   11. Simplified 11.8%

       \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y} + \left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1} - \sqrt{y}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + t\_2}\right) + \left(t\_3 - \sqrt{z}\right)\right) + t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(t\_1 + \left(t\_2 - \sqrt{x}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + t\_3}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ y 1.0)) (sqrt y)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= t_1 5e-5)
     (+
      (+ (fma 0.5 (sqrt (/ 1.0 y)) (/ 1.0 (+ (sqrt x) t_2))) (- t_3 (sqrt z)))
      t_4)
     (+
      t_4
      (+ (+ t_1 (- t_2 (sqrt x))) (/ (- (+ 1.0 z) z) (+ (sqrt z) t_3)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0)) - sqrt(y);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + z));
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (fma(0.5, sqrt((1.0 / y)), (1.0 / (sqrt(x) + t_2))) + (t_3 - sqrt(z))) + t_4;
	} else {
		tmp = t_4 + ((t_1 + (t_2 - sqrt(x))) + (((1.0 + z) - z) / (sqrt(z) + t_3)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(y + 1.0)) - sqrt(y))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = sqrt(Float64(1.0 + z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / y)), Float64(1.0 / Float64(sqrt(x) + t_2))) + Float64(t_3 - sqrt(z))) + t_4);
	else
		tmp = Float64(t_4 + Float64(Float64(t_1 + Float64(t_2 - sqrt(x))) + Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_3))));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], N[(t$95$4 + N[(N[(t$95$1 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 5.00000000000000024e-5

   1. Initial program 89.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(1 + y\right)} - y}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 89.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 89.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 89.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(1 + y\right) - y}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 91.7%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(1 + x\right) - x, \sqrt{1 + y} + \sqrt{y}, \left(\sqrt{x} + \sqrt{1 + x}\right) \cdot 1\right)}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \left(\sqrt{1 + y} + \sqrt{y}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 91.9

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 91.9%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-+.f64 96.8

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 96.8%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 22.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{y} + \left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.5e+31)
   (+ (sqrt y) (- (sqrt (+ 1.0 x)) (+ (sqrt y) (sqrt x))))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+31) {
		tmp = sqrt(y) + (sqrt((1.0 + x)) - (sqrt(y) + sqrt(x)));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 9.5d+31) then
        tmp = sqrt(y) + (sqrt((1.0d0 + x)) - (sqrt(y) + sqrt(x)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.5e+31) {
		tmp = Math.sqrt(y) + (Math.sqrt((1.0 + x)) - (Math.sqrt(y) + Math.sqrt(x)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 9.5e+31:
		tmp = math.sqrt(y) + (math.sqrt((1.0 + x)) - (math.sqrt(y) + math.sqrt(x)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.5e+31)
		tmp = Float64(sqrt(y) + Float64(sqrt(Float64(1.0 + x)) - Float64(sqrt(y) + sqrt(x))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9.5e+31)
		tmp = sqrt(y) + (sqrt((1.0 + x)) - (sqrt(y) + sqrt(x)));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, 9.5e+31], N[(N[Sqrt[y], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5000000000000008e31

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 34.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 33.7

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 33.7%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 12.4

          \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   11. Simplified 12.4%

       \[\leadsto \color{blue}{\sqrt{y}} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

   if 9.5000000000000008e31 < y

   1. Initial program 89.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 44.2

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 44.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 6.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{y} + \left(\sqrt{1 + x} - \left(\sqrt{y} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 16.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + z} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 4.8e-80)
   (+ (sqrt (+ 1.0 x)) (- (sqrt (+ 1.0 z)) (sqrt x)))
   (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.8e-80) {
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + z)) - sqrt(x));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 4.8d-80) then
        tmp = sqrt((1.0d0 + x)) + (sqrt((1.0d0 + z)) - sqrt(x))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 4.8e-80) {
		tmp = Math.sqrt((1.0 + x)) + (Math.sqrt((1.0 + z)) - Math.sqrt(x));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 4.8e-80:
		tmp = math.sqrt((1.0 + x)) + (math.sqrt((1.0 + z)) - math.sqrt(x))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 4.8e-80)
		tmp = Float64(sqrt(Float64(1.0 + x)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(x)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 4.8e-80)
		tmp = sqrt((1.0 + x)) + (sqrt((1.0 + z)) - sqrt(x));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 4.8e-80], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.7999999999999998e-80

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 17.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 17.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 16.9

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 16.9%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + z}\right) - \sqrt{x}} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} - \sqrt{x}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} - \sqrt{x}\right)} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{1 + x}} + \left(\sqrt{1 + z} - \sqrt{x}\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \sqrt{\color{blue}{1 + x}} + \left(\sqrt{1 + z} - \sqrt{x}\right) \]

       5. lower--.f64 N/A

          \[\leadsto \sqrt{1 + x} + \color{blue}{\left(\sqrt{1 + z} - \sqrt{x}\right)} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\color{blue}{\sqrt{1 + z}} - \sqrt{x}\right) \]

       7. lower-+.f64 N/A

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{\color{blue}{1 + z}} - \sqrt{x}\right) \]

       8. lower-sqrt.f64 20.1

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + z} - \color{blue}{\sqrt{x}}\right) \]

   11. Simplified 20.1%

       \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + z} - \sqrt{x}\right)} \]

   if 4.7999999999999998e-80 < x

   1. Initial program 90.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 73.0

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 73.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 7.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 12.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.75e-109) (* 0.5 (sqrt (/ 1.0 z))) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.75e-109) {
		tmp = 0.5 * sqrt((1.0 / z));
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.75d-109) then
        tmp = 0.5d0 * sqrt((1.0d0 / z))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.75e-109) {
		tmp = 0.5 * Math.sqrt((1.0 / z));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.75e-109:
		tmp = 0.5 * math.sqrt((1.0 / z))
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.75e-109)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / z)));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.75e-109)
		tmp = 0.5 * sqrt((1.0 / z));
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.75e-109], N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75e-109

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 59.0

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 59.0%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}} \]

      3. lower-/.f64 7.0

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{z}}} \]

   8. Simplified 7.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{z}}} \]

   if 1.75e-109 < x

   1. Initial program 91.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 69.2

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 69.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 7.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 12.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.75e-109) (sqrt z) (* 0.5 (sqrt (/ 1.0 x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.75e-109) {
		tmp = sqrt(z);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 1.75d-109) then
        tmp = sqrt(z)
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.75e-109) {
		tmp = Math.sqrt(z);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.75e-109:
		tmp = math.sqrt(z)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.75e-109)
		tmp = sqrt(z);
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.75e-109)
		tmp = sqrt(z);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[x, 1.75e-109], N[Sqrt[z], $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75e-109

   1. Initial program 96.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      16. lower-sqrt.f64 16.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 16.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 17.5

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 17.5%

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\sqrt{z}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 6.9

          \[\leadsto \color{blue}{\sqrt{z}} \]

   11. Simplified 6.9%

       \[\leadsto \color{blue}{\sqrt{z}} \]

   if 1.75e-109 < x

   1. Initial program 91.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \sqrt{1 + y}\right)} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y} - \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{x}}, \sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 69.2

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 69.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{x}}, \sqrt{1 + y} - \sqrt{y}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

      3. lower-/.f64 7.8

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}} \]

   8. Simplified 7.8%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 7.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{z} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt z))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(z);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(z)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(z);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(z)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(z)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(z);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[z], $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 20.0

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 20.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
7. Step-by-step derivation

   1. lower-sqrt.f64 20.1

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

8. Simplified 20.1%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\sqrt{z}} \]
10. Step-by-step derivation

    1. lower-sqrt.f64 6.9

       \[\leadsto \color{blue}{\sqrt{z}} \]

11. Simplified 6.9%

    \[\leadsto \color{blue}{\sqrt{z}} \]
12. Add Preprocessing

Alternative 22: 7.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (sqrt y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return sqrt(y);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt(y)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.sqrt(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return sqrt(y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt(y);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 92.9%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \color{blue}{\left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\color{blue}{\sqrt{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{\color{blue}{1 + x}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   16. lower-sqrt.f64 20.0

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 20.0%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{z} + \sqrt{y}\right)}\right)\right) \]

   4. flip-+ N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{z} \cdot \sqrt{z} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{\sqrt{z} \cdot \sqrt{z} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}}\right)\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{\color{blue}{\sqrt{z}} \cdot \sqrt{z} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{\sqrt{z} \cdot \color{blue}{\sqrt{z}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{\color{blue}{z} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{z - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{z - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   11. rem-square-sqrt N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{z - \color{blue}{y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{\color{blue}{z - y}}{\sqrt{z} - \sqrt{y}}\right)\right) \]

   13. lower--.f64 20.2

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \frac{z - y}{\color{blue}{\sqrt{z} - \sqrt{y}}}\right)\right) \]

7. Applied egg-rr 20.2%

   \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \left(\sqrt{1 + x} - \left(\sqrt{x} + \color{blue}{\frac{z - y}{\sqrt{z} - \sqrt{y}}}\right)\right) \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\sqrt{y}} \]
9. Step-by-step derivation

   1. lower-sqrt.f64 6.7

      \[\leadsto \color{blue}{\sqrt{y}} \]

10. Simplified 6.7%

    \[\leadsto \color{blue}{\sqrt{y}} \]
11. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024212 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))