Main:z from

Percentage Accurate: 91.9% → 99.4%

Time: 26.1s

Alternatives: 20

Speedup: 1.1×

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: 91.9% 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: 99.4% accurate, 0.7× speedup

Math

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

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 (+ z 1.0)) (sqrt z)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (sqrt (+ 1.0 y))))
   (if (<= t_2 2e-5)
     (+
      (+
       (fma 0.5 (sqrt (/ 1.0 z)) (/ 1.0 (+ (sqrt x) t_3)))
       (/ 1.0 (+ (sqrt y) t_4)))
      (- t_1 (sqrt t)))
     (+
      (+ t_2 (+ (- t_3 (sqrt x)) (- t_4 (sqrt y))))
      (/ (- (+ 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((z + 1.0)) - sqrt(z);
	double t_3 = sqrt((1.0 + x));
	double t_4 = sqrt((1.0 + y));
	double tmp;
	if (t_2 <= 2e-5) {
		tmp = (fma(0.5, sqrt((1.0 / z)), (1.0 / (sqrt(x) + t_3))) + (1.0 / (sqrt(y) + t_4))) + (t_1 - sqrt(t));
	} else {
		tmp = (t_2 + ((t_3 - sqrt(x)) + (t_4 - sqrt(y)))) + (((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 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (t_2 <= 2e-5)
		tmp = Float64(Float64(fma(0.5, sqrt(Float64(1.0 / z)), Float64(1.0 / Float64(sqrt(x) + t_3))) + Float64(1.0 / Float64(sqrt(y) + t_4))) + Float64(t_1 - sqrt(t)));
	else
		tmp = Float64(Float64(t_2 + Float64(Float64(t_3 - sqrt(x)) + Float64(t_4 - sqrt(y)))) + 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-5], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $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 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 2.00000000000000016e-5

   1. Initial program 88.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 89.2

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

   4. Applied egg-rr 89.2%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lift-+.f64 N/A

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

   6. Applied egg-rr 90.7%

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

   7. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 98.0

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

   9. Simplified 98.0%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 97.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.6

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

   4. Applied egg-rr 97.6%

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

4. Final simplification 97.9%

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

Alternative 2: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t}\\ t_3 := t\_2 - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := \sqrt{z + 1}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_4 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_6 \leq 1.0001:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_4\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_5\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_2\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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)))
        (t_3 (- t_2 (sqrt t)))
        (t_4 (sqrt (+ 1.0 y)))
        (t_5 (sqrt (+ z 1.0)))
        (t_6
         (+ t_3 (+ (- t_5 (sqrt z)) (+ (- t_1 (sqrt x)) (- t_4 (sqrt y)))))))
   (if (<= t_6 1.0001)
     (+ t_3 (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x)))
     (if (<= t_6 2.0002)
       (+ 1.0 (- (fma 0.5 (sqrt (/ 1.0 z)) t_4) (+ (sqrt x) (sqrt y))))
       (if (<= t_6 3.0)
         (- (+ 2.0 (fma y 0.5 t_5)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
         (+ 3.0 (- (fma x 0.5 t_2) (sqrt t))))))))

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));
	double t_3 = t_2 - sqrt(t);
	double t_4 = sqrt((1.0 + y));
	double t_5 = sqrt((z + 1.0));
	double t_6 = t_3 + ((t_5 - sqrt(z)) + ((t_1 - sqrt(x)) + (t_4 - sqrt(y))));
	double tmp;
	if (t_6 <= 1.0001) {
		tmp = t_3 + (fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x));
	} else if (t_6 <= 2.0002) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), t_4) - (sqrt(x) + sqrt(y)));
	} else if (t_6 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_5)) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_2) - sqrt(t));
	}
	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 + t))
	t_3 = Float64(t_2 - sqrt(t))
	t_4 = sqrt(Float64(1.0 + y))
	t_5 = sqrt(Float64(z + 1.0))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(z)) + Float64(Float64(t_1 - sqrt(x)) + Float64(t_4 - sqrt(y)))))
	tmp = 0.0
	if (t_6 <= 1.0001)
		tmp = Float64(t_3 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x)));
	elseif (t_6 <= 2.0002)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_4) - Float64(sqrt(x) + sqrt(y))));
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_5)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_2) - sqrt(t)));
	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 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0001], N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0002], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$2), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 81.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 39.7

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

   8. Simplified 39.7%

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

   if 1.00009999999999999 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 25.8

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

   5. Simplified 25.8%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 13.6

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

   8. Simplified 13.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.3

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

   11. Simplified 29.3%

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

   if 2.00019999999999998 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 38.1

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

   5. Simplified 38.1%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 37.5

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

   8. Simplified 37.5%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.1

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Simplified 29.1%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   8. Simplified 88.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

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

       1. lower-sqrt.f64 78.3

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 78.3%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 78.6

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 78.6%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{z + 1}\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := t\_1 - \sqrt{t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{z + 1}\\ t_5 := t\_2 + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_5 \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_3\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_4\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_1\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 t)))
        (t_2 (- t_1 (sqrt t)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5
         (+
          t_2
          (+
           (- t_4 (sqrt z))
           (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_3 (sqrt y)))))))
   (if (<= t_5 1.0001)
     (+ t_2 (- (fma 0.5 (+ x (sqrt (/ 1.0 y))) 1.0) (sqrt x)))
     (if (<= t_5 2.0002)
       (+ 1.0 (- (fma 0.5 (sqrt (/ 1.0 z)) t_3) (+ (sqrt x) (sqrt y))))
       (if (<= t_5 3.0)
         (- (+ 2.0 (fma y 0.5 t_4)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
         (+ 3.0 (- (fma x 0.5 t_1) (sqrt t))))))))

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 = t_1 - sqrt(t);
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((z + 1.0));
	double t_5 = t_2 + ((t_4 - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_3 - sqrt(y))));
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = t_2 + (fma(0.5, (x + sqrt((1.0 / y))), 1.0) - sqrt(x));
	} else if (t_5 <= 2.0002) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), t_3) - (sqrt(x) + sqrt(y)));
	} else if (t_5 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_4)) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_1) - sqrt(t));
	}
	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 = Float64(t_1 - sqrt(t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(t_2 + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(t_2 + Float64(fma(0.5, Float64(x + sqrt(Float64(1.0 / y))), 1.0) - sqrt(x)));
	elseif (t_5 <= 2.0002)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_3) - Float64(sqrt(x) + sqrt(y))));
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_4)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_1) - sqrt(t)));
	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[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(t$95$2 + N[(N[(0.5 * N[(x + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0002], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$1), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.00009999999999999

   1. Initial program 81.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 19.0

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

   5. Simplified 19.0%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 18.9

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

   8. Simplified 18.9%

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

   9. Taylor expanded in z around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sqrt.f64 18.6

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

   11. Simplified 18.6%

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

   if 1.00009999999999999 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 98.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 25.8

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

   5. Simplified 25.8%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 13.6

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

   8. Simplified 13.6%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.3

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

   11. Simplified 29.3%

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

   if 2.00019999999999998 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 38.1

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

   5. Simplified 38.1%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 37.5

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

   8. Simplified 37.5%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.1

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Simplified 29.1%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   8. Simplified 88.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

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

       1. lower-sqrt.f64 78.3

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 78.3%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 78.6

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 78.6%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, x + \sqrt{\frac{1}{y}}, 1\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{z + 1}\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t} - \sqrt{t}\\ t_5 := \sqrt{1 + x}\\ t_6 := t\_3 + \left(\left(t\_5 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0.0001:\\ \;\;\;\;t\_4 + \left(t\_3 + \frac{1}{\sqrt{x} + t\_5}\right)\\ \mathbf{elif}\;t\_6 \leq 1.0001:\\ \;\;\;\;t\_4 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_5\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 2.999998:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + t\_2} + \mathsf{fma}\left(x, 0.5, t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4 + \left(\left(t\_2 + \left(1 + t\_5\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_5 (sqrt (+ 1.0 x)))
        (t_6 (+ t_3 (+ (- t_5 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_6 0.0001)
     (+ t_4 (+ t_3 (/ 1.0 (+ (sqrt x) t_5))))
     (if (<= t_6 1.0001)
       (+ t_4 (- (fma 0.5 (sqrt (/ 1.0 y)) t_5) (sqrt x)))
       (if (<= t_6 2.999998)
         (+
          1.0
          (-
           (+ (/ 1.0 (+ (sqrt z) t_2)) (fma x 0.5 t_1))
           (+ (sqrt x) (sqrt y))))
         (+
          t_4
          (- (+ t_2 (+ 1.0 t_5)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t)) - sqrt(t);
	double t_5 = sqrt((1.0 + x));
	double t_6 = t_3 + ((t_5 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_6 <= 0.0001) {
		tmp = t_4 + (t_3 + (1.0 / (sqrt(x) + t_5)));
	} else if (t_6 <= 1.0001) {
		tmp = t_4 + (fma(0.5, sqrt((1.0 / y)), t_5) - sqrt(x));
	} else if (t_6 <= 2.999998) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = t_4 + ((t_2 + (1.0 + t_5)) - (sqrt(x) + (sqrt(z) + sqrt(y))));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_5 = sqrt(Float64(1.0 + x))
	t_6 = Float64(t_3 + Float64(Float64(t_5 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_6 <= 0.0001)
		tmp = Float64(t_4 + Float64(t_3 + Float64(1.0 / Float64(sqrt(x) + t_5))));
	elseif (t_6 <= 1.0001)
		tmp = Float64(t_4 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_5) - sqrt(x)));
	elseif (t_6 <= 2.999998)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(t_4 + Float64(Float64(t_2 + Float64(1.0 + t_5)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + N[(N[(t$95$5 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0001], N[(t$95$4 + N[(t$95$3 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.0001], N[(t$95$4 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.999998], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(t$95$2 + N[(1.0 + t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $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))) < 1.00000000000000005e-4

   1. Initial program 62.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 64.2

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

   4. Applied egg-rr 64.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. 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 79.4

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

   7. Simplified 79.4%

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

   if 1.00000000000000005e-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))) < 1.00009999999999999

   1. Initial program 96.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 56.6

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

   5. Simplified 56.6%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 34.8

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

   8. Simplified 34.8%

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

   if 1.00009999999999999 < (+.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))) < 2.99999800000000016

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 71.7

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

   5. Simplified 71.7%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower--.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 71.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 71.7

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

   7. Applied egg-rr 71.7%

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

   8. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   10. Simplified 34.7%

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

   if 2.99999800000000016 < (+.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 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. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.999998:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} + \left(1 + \sqrt{1 + x}\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := t\_2 - \sqrt{z}\\ t_4 := \sqrt{1 + t}\\ t_5 := t\_4 - \sqrt{t}\\ t_6 := \sqrt{1 + x}\\ t_7 := t\_3 + \left(\left(t\_6 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_7 \leq 0.0001:\\ \;\;\;\;t\_5 + \left(t\_3 + \frac{1}{\sqrt{x} + t\_6}\right)\\ \mathbf{elif}\;t\_7 \leq 1.0001:\\ \;\;\;\;t\_5 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_6\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_7 \leq 2.999999999999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + t\_2} + \mathsf{fma}\left(x, 0.5, t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_4\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- t_2 (sqrt z)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- t_4 (sqrt t)))
        (t_6 (sqrt (+ 1.0 x)))
        (t_7 (+ t_3 (+ (- t_6 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_7 0.0001)
     (+ t_5 (+ t_3 (/ 1.0 (+ (sqrt x) t_6))))
     (if (<= t_7 1.0001)
       (+ t_5 (- (fma 0.5 (sqrt (/ 1.0 y)) t_6) (sqrt x)))
       (if (<= t_7 2.999999999999995)
         (+
          1.0
          (-
           (+ (/ 1.0 (+ (sqrt z) t_2)) (fma x 0.5 t_1))
           (+ (sqrt x) (sqrt y))))
         (+ 3.0 (- (fma x 0.5 t_4) (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = t_2 - sqrt(z);
	double t_4 = sqrt((1.0 + t));
	double t_5 = t_4 - sqrt(t);
	double t_6 = sqrt((1.0 + x));
	double t_7 = t_3 + ((t_6 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_7 <= 0.0001) {
		tmp = t_5 + (t_3 + (1.0 / (sqrt(x) + t_6)));
	} else if (t_7 <= 1.0001) {
		tmp = t_5 + (fma(0.5, sqrt((1.0 / y)), t_6) - sqrt(x));
	} else if (t_7 <= 2.999999999999995) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_4) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(t_2 - sqrt(z))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(t_4 - sqrt(t))
	t_6 = sqrt(Float64(1.0 + x))
	t_7 = Float64(t_3 + Float64(Float64(t_6 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_7 <= 0.0001)
		tmp = Float64(t_5 + Float64(t_3 + Float64(1.0 / Float64(sqrt(x) + t_6))));
	elseif (t_7 <= 1.0001)
		tmp = Float64(t_5 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_6) - sqrt(x)));
	elseif (t_7 <= 2.999999999999995)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_4) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$3 + N[(N[(t$95$6 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0001], N[(t$95$5 + N[(t$95$3 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.0001], N[(t$95$5 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 2.999999999999995], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$4), $MachinePrecision] - N[Sqrt[t], $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))) < 1.00000000000000005e-4

   1. Initial program 62.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 64.2

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

   4. Applied egg-rr 64.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. 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 79.4

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

   7. Simplified 79.4%

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

   if 1.00000000000000005e-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))) < 1.00009999999999999

   1. Initial program 96.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 56.6

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

   5. Simplified 56.6%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 34.8

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

   8. Simplified 34.8%

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

   if 1.00009999999999999 < (+.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))) < 2.99999999999999512

   1. Initial program 97.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 73.4

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

   5. Simplified 73.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower--.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 73.4

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 73.4

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

   7. Applied egg-rr 73.4%

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

   8. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   10. Simplified 35.5%

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

   if 2.99999999999999512 < (+.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 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

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

       1. lower-sqrt.f64 46.7

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 46.7%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 99.9

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.999999999999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + t}\\ t_2 := \sqrt{1 + y}\\ t_3 := \sqrt{z + 1}\\ t_4 := \left(t\_1 - \sqrt{t}\right) + \left(\left(t\_3 - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_2 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_4 \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, t\_2\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_3\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_1\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 t)))
        (t_2 (sqrt (+ 1.0 y)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4
         (+
          (- t_1 (sqrt t))
          (+
           (- t_3 (sqrt z))
           (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_2 (sqrt y)))))))
   (if (<= t_4 2.0002)
     (+ 1.0 (- (fma 0.5 (sqrt (/ 1.0 z)) t_2) (+ (sqrt x) (sqrt y))))
     (if (<= t_4 3.0)
       (- (+ 2.0 (fma y 0.5 t_3)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
       (+ 3.0 (- (fma x 0.5 t_1) (sqrt t)))))))

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 + y));
	double t_3 = sqrt((z + 1.0));
	double t_4 = (t_1 - sqrt(t)) + ((t_3 - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_2 - sqrt(y))));
	double tmp;
	if (t_4 <= 2.0002) {
		tmp = 1.0 + (fma(0.5, sqrt((1.0 / z)), t_2) - (sqrt(x) + sqrt(y)));
	} else if (t_4 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_3)) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_1) - sqrt(t));
	}
	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 + y))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(Float64(t_1 - sqrt(t)) + Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_2 - sqrt(y)))))
	tmp = 0.0
	if (t_4 <= 2.0002)
		tmp = Float64(1.0 + Float64(fma(0.5, sqrt(Float64(1.0 / z)), t_2) - Float64(sqrt(x) + sqrt(y))));
	elseif (t_4 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_3)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_1) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 2.0002], N[(1.0 + N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$1), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.00019999999999998

   1. Initial program 90.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 15.9

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

   5. Simplified 15.9%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 26.4

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

   8. Simplified 26.4%

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

   9. Taylor expanded in z around inf

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

       1. associate--l+ N/A

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

       2. lower-+.f64 N/A

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

       3. lower--.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-+.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 28.9

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

   11. Simplified 28.9%

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

   if 2.00019999999999998 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 38.1

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

   5. Simplified 38.1%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 37.5

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

   8. Simplified 37.5%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 29.1

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Simplified 29.1%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   8. Simplified 88.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

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

       1. lower-sqrt.f64 78.3

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 78.3%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 78.6

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 78.6%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.0002:\\ \;\;\;\;1 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{z}}, \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{z + 1}\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{z + 1}\\ t_5 := \left(t\_2 - \sqrt{t}\right) + \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 2:\\ \;\;\;\;t\_1 + \left(t\_3 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, t\_4\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_2\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5
         (+
          (- t_2 (sqrt t))
          (+ (- t_4 (sqrt z)) (+ (- t_1 (sqrt x)) (- t_3 (sqrt y)))))))
   (if (<= t_5 2.0)
     (+ t_1 (- t_3 (+ (sqrt x) (sqrt y))))
     (if (<= t_5 3.0)
       (- (+ 2.0 (fma y 0.5 t_4)) (+ (sqrt x) (+ (sqrt z) (sqrt y))))
       (+ 3.0 (- (fma x 0.5 t_2) (sqrt t)))))))

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));
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((z + 1.0));
	double t_5 = (t_2 - sqrt(t)) + ((t_4 - sqrt(z)) + ((t_1 - sqrt(x)) + (t_3 - sqrt(y))));
	double tmp;
	if (t_5 <= 2.0) {
		tmp = t_1 + (t_3 - (sqrt(x) + sqrt(y)));
	} else if (t_5 <= 3.0) {
		tmp = (2.0 + fma(y, 0.5, t_4)) - (sqrt(x) + (sqrt(z) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_2) - sqrt(t));
	}
	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 + t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(t)) + Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_5 <= 2.0)
		tmp = Float64(t_1 + Float64(t_3 - Float64(sqrt(x) + sqrt(y))));
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(2.0 + fma(y, 0.5, t_4)) - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_2) - sqrt(t)));
	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 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 2.0], N[(t$95$1 + N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(2.0 + N[(y * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$2), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 90.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 15.7

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

   5. Simplified 15.7%

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

   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. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 25.1

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

   8. Simplified 25.1%

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

   if 2 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 37.8

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

   5. Simplified 37.8%

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

   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. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 36.5

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

   8. Simplified 36.5%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-+.f64 N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-sqrt.f64 N/A

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

       12. lower-sqrt.f64 28.2

           \[\leadsto \left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right) \]

   11. Simplified 28.2%

       \[\leadsto \color{blue}{\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

   if 3 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 94.3

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

   5. Simplified 94.3%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

   8. Simplified 88.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

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

       1. lower-sqrt.f64 78.3

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 78.3%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 78.6

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 78.6%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 3:\\ \;\;\;\;\left(2 + \mathsf{fma}\left(y, 0.5, \sqrt{z + 1}\right)\right) - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t}\\ t_3 := \sqrt{1 + y}\\ t_4 := \sqrt{z + 1} - \sqrt{z}\\ t_5 := \left(t\_2 - \sqrt{t}\right) + \left(t\_4 + \left(\left(t\_1 - \sqrt{x}\right) + \left(t\_3 - \sqrt{y}\right)\right)\right)\\ \mathbf{if}\;t\_5 \leq 1.9999999999999916:\\ \;\;\;\;t\_1 + \left(t\_3 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 2.999999999999995:\\ \;\;\;\;t\_4 + \left(t\_1 + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_2\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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)))
        (t_3 (sqrt (+ 1.0 y)))
        (t_4 (- (sqrt (+ z 1.0)) (sqrt z)))
        (t_5
         (+ (- t_2 (sqrt t)) (+ t_4 (+ (- t_1 (sqrt x)) (- t_3 (sqrt y)))))))
   (if (<= t_5 1.9999999999999916)
     (+ t_1 (- t_3 (+ (sqrt x) (sqrt y))))
     (if (<= t_5 2.999999999999995)
       (+ t_4 (+ t_1 t_3))
       (+ 3.0 (- (fma x 0.5 t_2) (sqrt t)))))))

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));
	double t_3 = sqrt((1.0 + y));
	double t_4 = sqrt((z + 1.0)) - sqrt(z);
	double t_5 = (t_2 - sqrt(t)) + (t_4 + ((t_1 - sqrt(x)) + (t_3 - sqrt(y))));
	double tmp;
	if (t_5 <= 1.9999999999999916) {
		tmp = t_1 + (t_3 - (sqrt(x) + sqrt(y)));
	} else if (t_5 <= 2.999999999999995) {
		tmp = t_4 + (t_1 + t_3);
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_2) - sqrt(t));
	}
	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 + t))
	t_3 = sqrt(Float64(1.0 + y))
	t_4 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_5 = Float64(Float64(t_2 - sqrt(t)) + Float64(t_4 + Float64(Float64(t_1 - sqrt(x)) + Float64(t_3 - sqrt(y)))))
	tmp = 0.0
	if (t_5 <= 1.9999999999999916)
		tmp = Float64(t_1 + Float64(t_3 - Float64(sqrt(x) + sqrt(y))));
	elseif (t_5 <= 2.999999999999995)
		tmp = Float64(t_4 + Float64(t_1 + t_3));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_2) - sqrt(t)));
	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 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.9999999999999916], N[(t$95$1 + N[(t$95$3 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.999999999999995], N[(t$95$4 + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$2), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1.9999999999999916

   1. Initial program 84.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 5.9

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

   5. Simplified 5.9%

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

   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. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 19.4

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

   8. Simplified 19.4%

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

   if 1.9999999999999916 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2.99999999999999512

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-sqrt.f64 30.6

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

   5. Simplified 30.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-sqrt.f64 28.2

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

   8. Simplified 28.2%

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

   if 2.99999999999999512 < (+.f64 (+.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))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 98.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 78.3

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 78.3%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 30.0

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 30.0%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 51.0

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 51.0%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 1.9999999999999916:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \leq 2.999999999999995:\\ \;\;\;\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{1 + x}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_5 \leq 1.0001:\\ \;\;\;\;\left(t\_3 - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_4\right) - \sqrt{x}\right)\\ \mathbf{elif}\;t\_5 \leq 2.999999999999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + t\_2} + \mathsf{fma}\left(x, 0.5, t\_1\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, t\_3\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ 1.0 x)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_5 1.0001)
     (+ (- t_3 (sqrt t)) (- (fma 0.5 (sqrt (/ 1.0 y)) t_4) (sqrt x)))
     (if (<= t_5 2.999999999999995)
       (+
        1.0
        (- (+ (/ 1.0 (+ (sqrt z) t_2)) (fma x 0.5 t_1)) (+ (sqrt x) (sqrt y))))
       (+ 3.0 (- (fma x 0.5 t_3) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((1.0 + x));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_5 <= 1.0001) {
		tmp = (t_3 - sqrt(t)) + (fma(0.5, sqrt((1.0 / y)), t_4) - sqrt(x));
	} else if (t_5 <= 2.999999999999995) {
		tmp = 1.0 + (((1.0 / (sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, t_3) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(1.0 + x))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_5 <= 1.0001)
		tmp = Float64(Float64(t_3 - sqrt(t)) + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_4) - sqrt(x)));
	elseif (t_5 <= 2.999999999999995)
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + fma(x, 0.5, t_1)) - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, t_3) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0001], N[(N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.999999999999995], N[(1.0 + N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + t$95$3), $MachinePrecision] - N[Sqrt[t], $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))) < 1.00009999999999999

   1. Initial program 87.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 59.8

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 59.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 41.4

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 41.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.00009999999999999 < (+.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))) < 2.99999999999999512

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 73.4

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 73.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      20. lower-+.f64 73.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      22. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      23. lift-+.f64 73.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 73.4%

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   10. Simplified 35.5%

       \[\leadsto \color{blue}{1 + \left(\left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 2.99999999999999512 < (+.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 99.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 99.9

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.9%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 46.7

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 46.7%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 99.9

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.0001:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.999999999999995:\\ \;\;\;\;1 + \left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + x}\\ t_4 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_3 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_4 \leq 1.9999999999999916:\\ \;\;\;\;t\_3 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2.999999999999995:\\ \;\;\;\;2 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ 1.0 x)))
        (t_4 (+ (- t_2 (sqrt z)) (+ (- t_3 (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_4 1.9999999999999916)
     (+ t_3 (- t_1 (+ (sqrt x) (sqrt y))))
     (if (<= t_4 2.999999999999995)
       (+ 2.0 (- t_2 (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
       (+ 3.0 (- (fma x 0.5 (sqrt (+ 1.0 t))) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((1.0 + x));
	double t_4 = (t_2 - sqrt(z)) + ((t_3 - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_4 <= 1.9999999999999916) {
		tmp = t_3 + (t_1 - (sqrt(x) + sqrt(y)));
	} else if (t_4 <= 2.999999999999995) {
		tmp = 2.0 + (t_2 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = 3.0 + (fma(x, 0.5, sqrt((1.0 + t))) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(1.0 + x))
	t_4 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_3 - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_4 <= 1.9999999999999916)
		tmp = Float64(t_3 + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	elseif (t_4 <= 2.999999999999995)
		tmp = Float64(2.0 + Float64(t_2 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, sqrt(Float64(1.0 + t))) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.9999999999999916], N[(t$95$3 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.999999999999995], N[(2.0 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $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))) < 1.9999999999999916

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 5.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. lower-+.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

      10. lower-sqrt.f64 17.0

          \[\leadsto \sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   8. Simplified 17.0%

      \[\leadsto \color{blue}{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 1.9999999999999916 < (+.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))) < 2.99999999999999512

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 34.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 34.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 18.4

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 18.4%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

       10. lower-sqrt.f64 28.9

           \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   11. Simplified 28.9%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 2.99999999999999512 < (+.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 99.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 99.9

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.9%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 46.7

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 46.7%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 99.9

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.9999999999999916:\\ \;\;\;\;\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.999999999999995:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \left(t\_2 - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\\ \mathbf{if}\;t\_3 \leq 1.9999999999999916:\\ \;\;\;\;1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2.999999999999995:\\ \;\;\;\;2 + \left(t\_2 - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3
         (+
          (- t_2 (sqrt z))
          (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y))))))
   (if (<= t_3 1.9999999999999916)
     (+ 1.0 (- t_1 (+ (sqrt x) (sqrt y))))
     (if (<= t_3 2.999999999999995)
       (+ 2.0 (- t_2 (+ (sqrt x) (+ (sqrt z) (sqrt y)))))
       (+ 3.0 (- (fma x 0.5 (sqrt (+ 1.0 t))) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = (t_2 - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y)));
	double tmp;
	if (t_3 <= 1.9999999999999916) {
		tmp = 1.0 + (t_1 - (sqrt(x) + sqrt(y)));
	} else if (t_3 <= 2.999999999999995) {
		tmp = 2.0 + (t_2 - (sqrt(x) + (sqrt(z) + sqrt(y))));
	} else {
		tmp = 3.0 + (fma(x, 0.5, sqrt((1.0 + t))) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y))))
	tmp = 0.0
	if (t_3 <= 1.9999999999999916)
		tmp = Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	elseif (t_3 <= 2.999999999999995)
		tmp = Float64(2.0 + Float64(t_2 - Float64(sqrt(x) + Float64(sqrt(z) + sqrt(y)))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, sqrt(Float64(1.0 + t))) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1.9999999999999916], N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.999999999999995], N[(2.0 + N[(t$95$2 - N[(N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $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))) < 1.9999999999999916

   1. Initial program 88.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 5.9

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.9%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 30.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 30.2%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       4. lower-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

       5. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

       6. lower-sqrt.f64 22.6

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   11. Simplified 22.6%

       \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   if 1.9999999999999916 < (+.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))) < 2.99999999999999512

   1. Initial program 97.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 34.0

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 34.0%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 18.4

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 18.4%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(2 + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 2 + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       5. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       6. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

       9. lower-sqrt.f64 N/A

          \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

       10. lower-sqrt.f64 28.9

           \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   11. Simplified 28.9%

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 2.99999999999999512 < (+.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 99.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 99.9

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 99.9%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 46.7%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 46.7

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 46.7%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 99.9

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 99.9%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.9999999999999916:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.999999999999995:\\ \;\;\;\;2 + \left(\sqrt{z + 1} - \left(\sqrt{x} + \left(\sqrt{z} + \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + x}\\ \mathbf{if}\;\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) \leq 1.999999:\\ \;\;\;\;t\_3 + \left(\frac{1}{\sqrt{x} + t\_4} + \frac{1}{\sqrt{y} + t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + t\_2}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \end{array} \]

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 y)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t)))
        (t_4 (sqrt (+ 1.0 x))))
   (if (<= (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt x)) (- t_1 (sqrt y)))) 1.999999)
     (+ t_3 (+ (/ 1.0 (+ (sqrt x) t_4)) (/ 1.0 (+ (sqrt y) t_1))))
     (+
      t_3
      (-
       (+ (fma x 0.5 2.0) (/ 1.0 (+ (sqrt z) t_2)))
       (+ (sqrt x) (sqrt y)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double t_4 = sqrt((1.0 + x));
	double tmp;
	if (((t_2 - sqrt(z)) + ((t_4 - sqrt(x)) + (t_1 - sqrt(y)))) <= 1.999999) {
		tmp = t_3 + ((1.0 / (sqrt(x) + t_4)) + (1.0 / (sqrt(y) + t_1)));
	} else {
		tmp = t_3 + ((fma(x, 0.5, 2.0) + (1.0 / (sqrt(z) + t_2))) - (sqrt(x) + sqrt(y)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_4 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(x)) + Float64(t_1 - sqrt(y)))) <= 1.999999)
		tmp = Float64(t_3 + Float64(Float64(1.0 / Float64(sqrt(x) + t_4)) + Float64(1.0 / Float64(sqrt(y) + t_1))));
	else
		tmp = Float64(t_3 + Float64(Float64(fma(x, 0.5, 2.0) + Float64(1.0 / Float64(sqrt(z) + t_2))) - Float64(sqrt(x) + sqrt(y))));
	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 + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.999999], N[(t$95$3 + N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(N[(x * 0.5 + 2.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 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))) < 1.99999900000000008

   1. Initial program 87.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 88.1

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.1%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{\color{blue}{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\color{blue}{\sqrt{y + 1}} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{\color{blue}{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{\color{blue}{1 + y}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{\color{blue}{1 + y}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\color{blue}{\sqrt{1 + y}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{1 + y} \cdot \sqrt{\color{blue}{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{1 + y} \cdot \sqrt{\color{blue}{1 + y}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{1 + y} \cdot \sqrt{\color{blue}{1 + y}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\sqrt{1 + y} \cdot \color{blue}{\sqrt{1 + y}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\color{blue}{\left(1 + y\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - y}{\sqrt{\color{blue}{1 + y}} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      20. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(1 + y\right) - y}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied egg-rr 90.2%

      \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{\sqrt{1 + y} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{\color{blue}{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\color{blue}{\sqrt{x}} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 78.0

          \[\leadsto \left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Simplified 78.0%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.99999900000000008 < (+.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 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 77.8

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 77.8%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      20. lower-+.f64 77.8

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      22. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      23. lift-+.f64 77.8

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 77.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(2 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 52.5

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 52.5%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 1.999999:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;1 + \left(t\_1 - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \end{array} \]

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 y))))
   (if (<=
        (+
         (- (sqrt (+ z 1.0)) (sqrt z))
         (+ (- (sqrt (+ 1.0 x)) (sqrt x)) (- t_1 (sqrt y))))
        2.5)
     (+ 1.0 (- t_1 (+ (sqrt x) (sqrt y))))
     (+ 3.0 (- (fma x 0.5 (sqrt (+ 1.0 t))) (sqrt t))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + y));
	double tmp;
	if (((sqrt((z + 1.0)) - sqrt(z)) + ((sqrt((1.0 + x)) - sqrt(x)) + (t_1 - sqrt(y)))) <= 2.5) {
		tmp = 1.0 + (t_1 - (sqrt(x) + sqrt(y)));
	} else {
		tmp = 3.0 + (fma(x, 0.5, sqrt((1.0 + t))) - sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + y))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(Float64(sqrt(Float64(1.0 + x)) - sqrt(x)) + Float64(t_1 - sqrt(y)))) <= 2.5)
		tmp = Float64(1.0 + Float64(t_1 - Float64(sqrt(x) + sqrt(y))));
	else
		tmp = Float64(3.0 + Float64(fma(x, 0.5, sqrt(Float64(1.0 + t))) - sqrt(t)));
	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 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.5], N[(1.0 + N[(t$95$1 - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(x * 0.5 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 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))) < 2.5

   1. Initial program 91.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 16.1

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 16.1%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 24.4

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 24.4%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\color{blue}{\sqrt{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{\color{blue}{1 + y}} - \left(\sqrt{x} + \sqrt{y}\right)\right) \]

       4. lower-+.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) \]

       5. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) \]

       6. lower-sqrt.f64 26.2

          \[\leadsto 1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

   11. Simplified 26.2%

       \[\leadsto 1 + \color{blue}{\left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)} \]

   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 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 x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 98.7

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 98.7%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 55.0%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 51.0

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 51.0%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 91.5

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 91.5%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) \leq 2.5:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1} - \sqrt{z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 1900000000:\\ \;\;\;\;t\_3 + \left(t\_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\mathsf{fma}\left(x, 0.5, 1\right) - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;t\_3 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_2\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \left(t\_1 + \frac{1}{\sqrt{x} + t\_2}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)) (sqrt z)))
        (t_2 (sqrt (+ 1.0 x)))
        (t_3 (- (sqrt (+ 1.0 t)) (sqrt t))))
   (if (<= y 1900000000.0)
     (+
      t_3
      (+ t_1 (+ (- (sqrt (+ 1.0 y)) (sqrt y)) (- (fma x 0.5 1.0) (sqrt x)))))
     (if (<= y 5.3e+29)
       (+ t_3 (- (fma 0.5 (sqrt (/ 1.0 y)) t_2) (sqrt x)))
       (+ t_3 (+ t_1 (/ 1.0 (+ (sqrt x) t_2))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0)) - sqrt(z);
	double t_2 = sqrt((1.0 + x));
	double t_3 = sqrt((1.0 + t)) - sqrt(t);
	double tmp;
	if (y <= 1900000000.0) {
		tmp = t_3 + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) + (fma(x, 0.5, 1.0) - sqrt(x))));
	} else if (y <= 5.3e+29) {
		tmp = t_3 + (fma(0.5, sqrt((1.0 / y)), t_2) - sqrt(x));
	} else {
		tmp = t_3 + (t_1 + (1.0 / (sqrt(x) + t_2)));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + 1.0)) - sqrt(z))
	t_2 = sqrt(Float64(1.0 + x))
	t_3 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	tmp = 0.0
	if (y <= 1900000000.0)
		tmp = Float64(t_3 + Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) + Float64(fma(x, 0.5, 1.0) - sqrt(x)))));
	elseif (y <= 5.3e+29)
		tmp = Float64(t_3 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_2) - sqrt(x)));
	else
		tmp = Float64(t_3 + Float64(t_1 + Float64(1.0 / Float64(sqrt(x) + 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[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1900000000.0], N[(t$95$3 + N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.5 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+29], N[(t$95$3 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(t$95$1 + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9e9

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\left(\color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 1\right) - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 52.0

         \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 52.0%

      \[\leadsto \left(\left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.9e9 < y < 5.3e29

   1. Initial program 77.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 98.2

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 98.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 39.3

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 39.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.3e29 < y

   1. Initial program 87.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 88.0

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.0%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. 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 93.6

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 93.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1900000000:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\mathsf{fma}\left(x, 0.5, 1\right) - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \sqrt{z + 1}\\ \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;t\_2 + \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + t\_3}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;t\_2 + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, t\_1\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(\left(t\_3 - \sqrt{z}\right) + \frac{1}{\sqrt{x} + t\_1}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0))))
   (if (<= y 0.68)
     (+
      t_2
      (- (+ (fma x 0.5 2.0) (/ 1.0 (+ (sqrt z) t_3))) (+ (sqrt x) (sqrt y))))
     (if (<= y 5.3e+29)
       (+ t_2 (- (fma 0.5 (sqrt (/ 1.0 y)) t_1) (sqrt x)))
       (+ t_2 (+ (- t_3 (sqrt z)) (/ 1.0 (+ (sqrt x) 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 + x));
	double t_2 = sqrt((1.0 + t)) - sqrt(t);
	double t_3 = sqrt((z + 1.0));
	double tmp;
	if (y <= 0.68) {
		tmp = t_2 + ((fma(x, 0.5, 2.0) + (1.0 / (sqrt(z) + t_3))) - (sqrt(x) + sqrt(y)));
	} else if (y <= 5.3e+29) {
		tmp = t_2 + (fma(0.5, sqrt((1.0 / y)), t_1) - sqrt(x));
	} else {
		tmp = t_2 + ((t_3 - sqrt(z)) + (1.0 / (sqrt(x) + 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 + x))
	t_2 = Float64(sqrt(Float64(1.0 + t)) - sqrt(t))
	t_3 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (y <= 0.68)
		tmp = Float64(t_2 + Float64(Float64(fma(x, 0.5, 2.0) + Float64(1.0 / Float64(sqrt(z) + t_3))) - Float64(sqrt(x) + sqrt(y))));
	elseif (y <= 5.3e+29)
		tmp = Float64(t_2 + Float64(fma(0.5, sqrt(Float64(1.0 / y)), t_1) - sqrt(x)));
	else
		tmp = Float64(t_2 + Float64(Float64(t_3 - sqrt(z)) + Float64(1.0 / Float64(sqrt(x) + 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 + 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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 0.68], N[(t$95$2 + N[(N[(N[(x * 0.5 + 2.0), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+29], N[(t$95$2 + N[(N[(0.5 * N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.680000000000000049

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 51.4

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{\color{blue}{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\color{blue}{\sqrt{z + 1}} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \color{blue}{\sqrt{z}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{1 + z} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      20. lower-+.f64 51.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      22. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      23. lift-+.f64 51.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied egg-rr 51.4%

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\left(2 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \left(\frac{1}{2} \cdot x + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(2 + \frac{1}{2} \cdot x\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 2\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 2\right)} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\color{blue}{\sqrt{z}} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \color{blue}{\sqrt{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\color{blue}{\sqrt{x}} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 51.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Simplified 51.4%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.680000000000000049 < y < 5.3e29

   1. Initial program 86.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x} - \sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 73.0

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 73.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x} - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\color{blue}{\frac{1}{y}}}, \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{\frac{1}{y}}, \sqrt{\color{blue}{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 31.1

         \[\leadsto \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Simplified 31.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.3e29 < y

   1. Initial program 87.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{x + 1}} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{x + 1} \cdot \color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 88.0

          \[\leadsto \left(\left(\frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied egg-rr 88.0%

      \[\leadsto \left(\left(\color{blue}{\frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. 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 93.6

         \[\leadsto \left(\frac{1}{\sqrt{x} + \sqrt{\color{blue}{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Simplified 93.6%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\mathsf{fma}\left(x, 0.5, 2\right) + \frac{1}{\sqrt{z} + \sqrt{z + 1}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+29}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}}, \sqrt{1 + x}\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \end{array} \]

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 0.85)
   (+ 3.0 (- (fma x 0.5 (sqrt (+ 1.0 t))) (sqrt t)))
   (- 1.0 (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 3.0 + (fma(x, 0.5, sqrt((1.0 + t))) - sqrt(t));
	} else {
		tmp = 1.0 - sqrt(x);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.85)
		tmp = Float64(3.0 + Float64(fma(x, 0.5, sqrt(Float64(1.0 + t))) - sqrt(t)));
	else
		tmp = Float64(1.0 - 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_] := If[LessEqual[y, 0.85], N[(3.0 + N[(N[(x * 0.5 + N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.849999999999999978

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 51.4

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 16.1

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 16.1%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right)\right) - \sqrt{t}} \]
   13. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{3 + \left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       3. lower--.f64 N/A

          \[\leadsto 3 + \color{blue}{\left(\left(\sqrt{1 + t} + \frac{1}{2} \cdot x\right) - \sqrt{t}\right)} \]

       4. +-commutative N/A

          \[\leadsto 3 + \left(\color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       5. *-commutative N/A

          \[\leadsto 3 + \left(\left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + t}\right) - \sqrt{t}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto 3 + \left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + t}\right)} - \sqrt{t}\right) \]

       7. lower-sqrt.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + t}}\right) - \sqrt{t}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + t}}\right) - \sqrt{t}\right) \]

       9. lower-sqrt.f64 28.0

          \[\leadsto 3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \color{blue}{\sqrt{t}}\right) \]

   14. Simplified 28.0%

       \[\leadsto \color{blue}{3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)} \]

   if 0.849999999999999978 < y

   1. Initial program 87.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 5.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 31.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

       3. lower-sqrt.f64 18.5

          \[\leadsto 1 + \left(-\color{blue}{\sqrt{x}}\right) \]

   11. Simplified 18.5%

       \[\leadsto 1 + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;3 + \left(\mathsf{fma}\left(x, 0.5, \sqrt{1 + t}\right) - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;2 + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \end{array} \]

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 0.85) (+ 2.0 (fma x 0.5 (sqrt (+ 1.0 y)))) (- 1.0 (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 0.85) {
		tmp = 2.0 + fma(x, 0.5, sqrt((1.0 + y)));
	} else {
		tmp = 1.0 - sqrt(x);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 0.85)
		tmp = Float64(2.0 + fma(x, 0.5, sqrt(Float64(1.0 + y))));
	else
		tmp = Float64(1.0 - 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_] := If[LessEqual[y, 0.85], N[(2.0 + N[(x * 0.5 + N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.849999999999999978

   1. Initial program 97.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 51.4

         \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Simplified 51.4%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   8. Simplified 14.1%

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f64 16.1

          \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   11. Simplified 16.1%

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

   12. Taylor expanded in t around inf

       \[\leadsto \color{blue}{2 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)} \]
   13. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \color{blue}{2 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)} \]

       2. +-commutative N/A

          \[\leadsto 2 + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)} \]

       3. *-commutative N/A

          \[\leadsto 2 + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto 2 + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)} \]

       5. lower-sqrt.f64 N/A

          \[\leadsto 2 + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) \]

       6. lower-+.f64 26.9

          \[\leadsto 2 + \mathsf{fma}\left(x, 0.5, \sqrt{\color{blue}{1 + y}}\right) \]

   14. Simplified 26.9%

       \[\leadsto \color{blue}{2 + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)} \]

   if 0.849999999999999978 < y

   1. Initial program 87.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

      17. lower-sqrt.f64 5.7

          \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   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. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

      14. lower-sqrt.f64 31.2

          \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto 1 + \color{blue}{-1 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

       3. lower-sqrt.f64 18.5

          \[\leadsto 1 + \left(-\color{blue}{\sqrt{x}}\right) \]

   11. Simplified 18.5%

       \[\leadsto 1 + \color{blue}{\left(-\sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.85:\\ \;\;\;\;2 + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- 1.0 (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0 - sqrt(x);
}

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 = 1.0d0 - sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 - Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0 - math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(1.0 - sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 - sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right)} + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + x}}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. lower--.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \color{blue}{\left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\color{blue}{\sqrt{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{\color{blue}{1 + z}} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\color{blue}{\sqrt{x}} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   15. lower-+.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]

   16. lower-sqrt.f64 N/A

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\color{blue}{\sqrt{y}} + \sqrt{z}\right)\right)\right) \]

   17. lower-sqrt.f64 21.1

       \[\leadsto \left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \color{blue}{\sqrt{z}}\right)\right)\right) \]

5. Simplified 21.1%

   \[\leadsto \color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + x}\right) + \left(\sqrt{1 + z} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

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. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\sqrt{1 + y} + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\sqrt{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{\color{blue}{1 + y}} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. associate-+r+ N/A

      \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

   10. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{x}} + \sqrt{y}\right) + \sqrt{z}\right)\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) + \sqrt{z}\right)\right) \]

   14. lower-sqrt.f64 28.4

       \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \color{blue}{\sqrt{z}}\right)\right) \]

8. Simplified 28.4%

   \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right)\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto 1 + \color{blue}{-1 \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

    2. lower-neg.f64 N/A

       \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\sqrt{x}\right)\right)} \]

    3. lower-sqrt.f64 13.7

       \[\leadsto 1 + \left(-\color{blue}{\sqrt{x}}\right) \]

11. Simplified 13.7%

    \[\leadsto 1 + \color{blue}{\left(-\sqrt{x}\right)} \]

12. Final simplification 13.7%

    \[\leadsto 1 - \sqrt{x} \]
13. Add Preprocessing

Alternative 19: 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 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate--l+ N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower--.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-sqrt.f64 52.6

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 52.6%

   \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

8. Simplified 9.5%

   \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

9. Taylor expanded in t around inf

   \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
10. Step-by-step derivation

    1. lower-sqrt.f64 11.1

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

11. Simplified 11.1%

    \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

12. Taylor expanded in y around inf

    \[\leadsto \color{blue}{\sqrt{y}} \]
13. Step-by-step derivation

    1. lower-sqrt.f64 6.8

       \[\leadsto \color{blue}{\sqrt{y}} \]

14. Simplified 6.8%

    \[\leadsto \color{blue}{\sqrt{y}} \]
15. Add Preprocessing

Alternative 20: 4.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot x \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* 0.5 x))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * x;
}

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 = 0.5d0 * x
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * x;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * x

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * x)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * x;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 92.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate--l+ N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{x \cdot \frac{1}{2}} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower--.f64 N/A

      \[\leadsto \left(\left(\mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-sqrt.f64 52.6

      \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Simplified 52.6%

   \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(2 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(2 + \sqrt{1 + t}\right)} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(2 + \color{blue}{\sqrt{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   6. lower-+.f64 N/A

      \[\leadsto \left(\left(2 + \sqrt{\color{blue}{1 + t}}\right) + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \left(\color{blue}{x \cdot \frac{1}{2}} + \sqrt{1 + y}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\left(\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) + \sqrt{t}\right)} \]

8. Simplified 9.5%

   \[\leadsto \color{blue}{\left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \left(\left(\left(\sqrt{x} + \sqrt{y}\right) + \sqrt{z}\right) + \sqrt{t}\right)} \]

9. Taylor expanded in t around inf

   \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, \frac{1}{2}, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]
10. Step-by-step derivation

    1. lower-sqrt.f64 11.1

       \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

11. Simplified 11.1%

    \[\leadsto \left(\left(2 + \sqrt{1 + t}\right) + \mathsf{fma}\left(x, 0.5, \sqrt{1 + y}\right)\right) - \color{blue}{\sqrt{t}} \]

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

    2. lower-*.f64 5.1

       \[\leadsto \color{blue}{x \cdot 0.5} \]

14. Simplified 5.1%

    \[\leadsto \color{blue}{x \cdot 0.5} \]

15. Final simplification 5.1%

    \[\leadsto 0.5 \cdot x \]
16. Add Preprocessing

Developer Target 1: 99.4% 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 2024214 
(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))))