Main:z from

Percentage Accurate: 91.8% → 98.7%

Time: 23.2s

Alternatives: 18

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := \sqrt{y + 1}\\ t_6 := \frac{1}{t\_5 + \sqrt{y}}\\ t_7 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_7 \leq 0:\\ \;\;\;\;t\_4 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_6\right)\\ \mathbf{elif}\;t\_7 \leq 1.8:\\ \;\;\;\;\left(\left(t\_6 + t\_1\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_7 \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{z}} + \left(t\_5 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + t\_3} + \left(t\_2 + 1\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 x)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (/ 1.0 (+ t_5 (sqrt y))))
        (t_7 (+ (- t_2 (sqrt z)) (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_7 0.0)
     (+ t_4 (fma (sqrt (/ 1.0 x)) 0.5 t_6))
     (if (<= t_7 1.8)
       (+ (- (+ t_6 t_1) (sqrt x)) t_4)
       (if (<= t_7 2.999999995)
         (- (+ (/ 1.0 (+ t_2 (sqrt z))) (+ t_5 1.0)) (+ (sqrt y) (sqrt x)))
         (+
          (-
           (+ (/ 1.0 (+ (sqrt t) t_3)) (+ t_2 1.0))
           (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
          1.0))))))

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((z + 1.0));
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = sqrt((y + 1.0));
	double t_6 = 1.0 / (t_5 + sqrt(y));
	double t_7 = (t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_7 <= 0.0) {
		tmp = t_4 + fma(sqrt((1.0 / x)), 0.5, t_6);
	} else if (t_7 <= 1.8) {
		tmp = ((t_6 + t_1) - sqrt(x)) + t_4;
	} else if (t_7 <= 2.999999995) {
		tmp = ((1.0 / (t_2 + sqrt(z))) + (t_5 + 1.0)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (((1.0 / (sqrt(t) + t_3)) + (t_2 + 1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	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(z + 1.0))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(1.0 / Float64(t_5 + sqrt(y)))
	t_7 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_7 <= 0.0)
		tmp = Float64(t_4 + fma(sqrt(Float64(1.0 / x)), 0.5, t_6));
	elseif (t_7 <= 1.8)
		tmp = Float64(Float64(Float64(t_6 + t_1) - sqrt(x)) + t_4);
	elseif (t_7 <= 2.999999995)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(t_5 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_3)) + Float64(t_2 + 1.0)) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0);
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0], N[(t$95$4 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.8], N[(N[(N[(t$95$6 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$7, 2.999999995], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 46.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.1%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.80000000000000004

   1. Initial program 95.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 39.3

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

   7. Applied rewrites 39.3%

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

   if 1.80000000000000004 < (+.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))) < 2.99999999500000003

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.7%

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

   if 2.99999999500000003 < (+.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 99.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \left(\sqrt{z + 1} + 1\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \frac{1}{t\_1 + \sqrt{y}}\\ t_3 := \sqrt{t + 1}\\ t_4 := \sqrt{z + 1}\\ t_5 := t\_3 - \sqrt{t}\\ t_6 := \left(\left(t\_4 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + t\_5\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;t\_5 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_2\right)\\ \mathbf{elif}\;t\_6 \leq 1.8:\\ \;\;\;\;\left(\left(t\_2 + 1\right) - \sqrt{x}\right) + t\_5\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(\frac{1}{t\_4 + \sqrt{z}} + \left(t\_1 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, t\_1\right) + t\_3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (/ 1.0 (+ t_1 (sqrt y))))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5 (- t_3 (sqrt t)))
        (t_6
         (+
          (+
           (- t_4 (sqrt z))
           (+ (- t_1 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))
          t_5)))
   (if (<= t_6 0.0)
     (+ t_5 (fma (sqrt (/ 1.0 x)) 0.5 t_2))
     (if (<= t_6 1.8)
       (+ (- (+ t_2 1.0) (sqrt x)) t_5)
       (if (<= t_6 3.0)
         (- (+ (/ 1.0 (+ t_4 (sqrt z))) (+ t_1 1.0)) (+ (sqrt y) (sqrt x)))
         (+
          (-
           (+ (fma 0.5 x t_1) t_3)
           (+ (+ (+ (sqrt z) (sqrt y)) (sqrt x)) (sqrt t)))
          2.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = 1.0 / (t_1 + sqrt(y));
	double t_3 = sqrt((t + 1.0));
	double t_4 = sqrt((z + 1.0));
	double t_5 = t_3 - sqrt(t);
	double t_6 = ((t_4 - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)))) + t_5;
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_5 + fma(sqrt((1.0 / x)), 0.5, t_2);
	} else if (t_6 <= 1.8) {
		tmp = ((t_2 + 1.0) - sqrt(x)) + t_5;
	} else if (t_6 <= 3.0) {
		tmp = ((1.0 / (t_4 + sqrt(z))) + (t_1 + 1.0)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((fma(0.5, x, t_1) + t_3) - (((sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 2.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = Float64(1.0 / Float64(t_1 + sqrt(y)))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(t_3 - sqrt(t))
	t_6 = Float64(Float64(Float64(t_4 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x)))) + t_5)
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(t_5 + fma(sqrt(Float64(1.0 / x)), 0.5, t_2));
	elseif (t_6 <= 1.8)
		tmp = Float64(Float64(Float64(t_2 + 1.0) - sqrt(x)) + t_5);
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(z))) + Float64(t_1 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(fma(0.5, x, t_1) + t_3) - Float64(Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 2.0);
	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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(t$95$4 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(t$95$5 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.8], N[(N[(N[(t$95$2 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * x + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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))) < 0.0

   1. Initial program 3.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 3.4

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

   4. Applied rewrites 3.4%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 3.4

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

   7. Applied rewrites 3.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 39.0%

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

   if 0.0 < (+.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))) < 1.80000000000000004

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if 1.80000000000000004 < (+.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 96.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 23.2%

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

   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 99.2%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.8%

      \[\leadsto 2 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{t + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := t\_2 - \sqrt{t}\\ t_5 := \left(\left(t\_3 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + t\_4\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + t\_4\\ \mathbf{elif}\;t\_5 \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{t\_1 + \sqrt{y}} + 1\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(\frac{1}{t\_3 + \sqrt{z}} + \left(t\_1 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, t\_1\right) + t\_2\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- t_2 (sqrt t)))
        (t_5
         (+
          (+
           (- t_3 (sqrt z))
           (+ (- t_1 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))
          t_4)))
   (if (<= t_5 0.0)
     (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 x))) 0.5) t_4)
     (if (<= t_5 1.8)
       (+ (- (+ (/ 1.0 (+ t_1 (sqrt y))) 1.0) (sqrt x)) t_4)
       (if (<= t_5 3.0)
         (- (+ (/ 1.0 (+ t_3 (sqrt z))) (+ t_1 1.0)) (+ (sqrt y) (sqrt x)))
         (+
          (-
           (+ (fma 0.5 x t_1) t_2)
           (+ (+ (+ (sqrt z) (sqrt y)) (sqrt x)) (sqrt t)))
          2.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((t + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = t_2 - sqrt(t);
	double t_5 = ((t_3 - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)))) + t_4;
	double tmp;
	if (t_5 <= 0.0) {
		tmp = ((sqrt((1.0 / z)) + sqrt((1.0 / x))) * 0.5) + t_4;
	} else if (t_5 <= 1.8) {
		tmp = (((1.0 / (t_1 + sqrt(y))) + 1.0) - sqrt(x)) + t_4;
	} else if (t_5 <= 3.0) {
		tmp = ((1.0 / (t_3 + sqrt(z))) + (t_1 + 1.0)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((fma(0.5, x, t_1) + t_2) - (((sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 2.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(t_2 - sqrt(t))
	t_5 = Float64(Float64(Float64(t_3 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x)))) + t_4)
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / x))) * 0.5) + t_4);
	elseif (t_5 <= 1.8)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + 1.0) - sqrt(x)) + t_4);
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(z))) + Float64(t_1 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(fma(0.5, x, t_1) + t_2) - Float64(Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)) + sqrt(t))) + 2.0);
	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[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 1.8], N[(N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * x + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 54.8%

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

   if 0.0 < (+.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))) < 1.80000000000000004

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if 1.80000000000000004 < (+.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 96.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 23.2%

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

   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 99.2%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.8%

      \[\leadsto 2 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 3:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1}\\ t_4 := t\_3 - \sqrt{t}\\ t_5 := \sqrt{y + 1}\\ t_6 := \frac{1}{t\_5 + \sqrt{y}}\\ t_7 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_7 \leq 0:\\ \;\;\;\;t\_4 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_6\right)\\ \mathbf{elif}\;t\_7 \leq 1.8:\\ \;\;\;\;\left(\left(t\_6 + t\_1\right) - \sqrt{x}\right) + t\_4\\ \mathbf{elif}\;t\_7 \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{z}} + \left(t\_5 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + t\_3} + t\_2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 2\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_3 (sqrt (+ t 1.0)))
        (t_4 (- t_3 (sqrt t)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (/ 1.0 (+ t_5 (sqrt y))))
        (t_7 (+ (- t_2 (sqrt z)) (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_7 0.0)
     (+ t_4 (fma (sqrt (/ 1.0 x)) 0.5 t_6))
     (if (<= t_7 1.8)
       (+ (- (+ t_6 t_1) (sqrt x)) t_4)
       (if (<= t_7 2.999999995)
         (- (+ (/ 1.0 (+ t_2 (sqrt z))) (+ t_5 1.0)) (+ (sqrt y) (sqrt x)))
         (+
          (-
           (+ (/ 1.0 (+ (sqrt t) t_3)) t_2)
           (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
          2.0))))))

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((z + 1.0));
	double t_3 = sqrt((t + 1.0));
	double t_4 = t_3 - sqrt(t);
	double t_5 = sqrt((y + 1.0));
	double t_6 = 1.0 / (t_5 + sqrt(y));
	double t_7 = (t_2 - sqrt(z)) + ((t_5 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_7 <= 0.0) {
		tmp = t_4 + fma(sqrt((1.0 / x)), 0.5, t_6);
	} else if (t_7 <= 1.8) {
		tmp = ((t_6 + t_1) - sqrt(x)) + t_4;
	} else if (t_7 <= 2.999999995) {
		tmp = ((1.0 / (t_2 + sqrt(z))) + (t_5 + 1.0)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (((1.0 / (sqrt(t) + t_3)) + t_2) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 2.0;
	}
	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(z + 1.0))
	t_3 = sqrt(Float64(t + 1.0))
	t_4 = Float64(t_3 - sqrt(t))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(1.0 / Float64(t_5 + sqrt(y)))
	t_7 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_7 <= 0.0)
		tmp = Float64(t_4 + fma(sqrt(Float64(1.0 / x)), 0.5, t_6));
	elseif (t_7 <= 1.8)
		tmp = Float64(Float64(Float64(t_6 + t_1) - sqrt(x)) + t_4);
	elseif (t_7 <= 2.999999995)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(t_5 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_3)) + t_2) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 2.0);
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(1.0 / N[(t$95$5 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0], N[(t$95$4 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1.8], N[(N[(N[(t$95$6 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$7, 2.999999995], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 46.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.1%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.80000000000000004

   1. Initial program 95.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 39.3

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

   7. Applied rewrites 39.3%

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

   if 1.80000000000000004 < (+.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))) < 2.99999999500000003

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.7%

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

   if 2.99999999500000003 < (+.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 99.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{t + 1}\\ t_3 := t\_2 - \sqrt{t}\\ t_4 := \sqrt{y + 1}\\ t_5 := \frac{1}{t\_4 + \sqrt{y}}\\ t_6 := \left(t\_1 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;t\_3 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_5\right)\\ \mathbf{elif}\;t\_6 \leq 1.8:\\ \;\;\;\;\left(\left(t\_5 + 1\right) - \sqrt{x}\right) + t\_3\\ \mathbf{elif}\;t\_6 \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{t\_1 + \sqrt{z}} + \left(t\_4 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + t\_2} + t\_1\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 2\\ \end{array} \end{array} \]

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)))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (- t_2 (sqrt t)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (/ 1.0 (+ t_4 (sqrt y))))
        (t_6
         (+
          (- t_1 (sqrt z))
          (+ (- t_4 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))))
   (if (<= t_6 0.0)
     (+ t_3 (fma (sqrt (/ 1.0 x)) 0.5 t_5))
     (if (<= t_6 1.8)
       (+ (- (+ t_5 1.0) (sqrt x)) t_3)
       (if (<= t_6 2.999999995)
         (- (+ (/ 1.0 (+ t_1 (sqrt z))) (+ t_4 1.0)) (+ (sqrt y) (sqrt x)))
         (+
          (-
           (+ (/ 1.0 (+ (sqrt t) t_2)) t_1)
           (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
          2.0))))))

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));
	double t_2 = sqrt((t + 1.0));
	double t_3 = t_2 - sqrt(t);
	double t_4 = sqrt((y + 1.0));
	double t_5 = 1.0 / (t_4 + sqrt(y));
	double t_6 = (t_1 - sqrt(z)) + ((t_4 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_3 + fma(sqrt((1.0 / x)), 0.5, t_5);
	} else if (t_6 <= 1.8) {
		tmp = ((t_5 + 1.0) - sqrt(x)) + t_3;
	} else if (t_6 <= 2.999999995) {
		tmp = ((1.0 / (t_1 + sqrt(z))) + (t_4 + 1.0)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (((1.0 / (sqrt(t) + t_2)) + t_1) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 2.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(t + 1.0))
	t_3 = Float64(t_2 - sqrt(t))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(1.0 / Float64(t_4 + sqrt(y)))
	t_6 = Float64(Float64(t_1 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x))))
	tmp = 0.0
	if (t_6 <= 0.0)
		tmp = Float64(t_3 + fma(sqrt(Float64(1.0 / x)), 0.5, t_5));
	elseif (t_6 <= 1.8)
		tmp = Float64(Float64(Float64(t_5 + 1.0) - sqrt(x)) + t_3);
	elseif (t_6 <= 2.999999995)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(z))) + Float64(t_4 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(t) + t_2)) + t_1) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x))) + 2.0);
	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[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(t$95$3 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 1.8], N[(N[(N[(t$95$5 + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$6, 2.999999995], N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[(N[Sqrt[t], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 46.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.1%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.80000000000000004

   1. Initial program 95.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 39.3

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 27.8%

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

   if 1.80000000000000004 < (+.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))) < 2.99999999500000003

   1. Initial program 96.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.7%

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

   if 2.99999999500000003 < (+.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 99.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.999999995:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \left(\left(t\_2 - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + t\_3\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + t\_3\\ \mathbf{elif}\;t\_4 \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{t\_1 + \sqrt{y}} + 1\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_2 + \sqrt{z}} + \left(t\_1 + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4
         (+
          (+
           (- t_2 (sqrt z))
           (+ (- t_1 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))
          t_3)))
   (if (<= t_4 0.0)
     (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 x))) 0.5) t_3)
     (if (<= t_4 1.8)
       (+ (- (+ (/ 1.0 (+ t_1 (sqrt y))) 1.0) (sqrt x)) t_3)
       (- (+ (/ 1.0 (+ t_2 (sqrt z))) (+ t_1 1.0)) (+ (sqrt y) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)))) + t_3;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = ((sqrt((1.0 / z)) + sqrt((1.0 / x))) * 0.5) + t_3;
	} else if (t_4 <= 1.8) {
		tmp = (((1.0 / (t_1 + sqrt(y))) + 1.0) - sqrt(x)) + t_3;
	} else {
		tmp = ((1.0 / (t_2 + sqrt(z))) + (t_1 + 1.0)) - (sqrt(y) + sqrt(x));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    t_4 = ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0d0 + x)) - sqrt(x)))) + t_3
    if (t_4 <= 0.0d0) then
        tmp = ((sqrt((1.0d0 / z)) + sqrt((1.0d0 / x))) * 0.5d0) + t_3
    else if (t_4 <= 1.8d0) then
        tmp = (((1.0d0 / (t_1 + sqrt(y))) + 1.0d0) - sqrt(x)) + t_3
    else
        tmp = ((1.0d0 / (t_2 + sqrt(z))) + (t_1 + 1.0d0)) - (sqrt(y) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_4 = ((t_2 - Math.sqrt(z)) + ((t_1 - Math.sqrt(y)) + (Math.sqrt((1.0 + x)) - Math.sqrt(x)))) + t_3;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = ((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / x))) * 0.5) + t_3;
	} else if (t_4 <= 1.8) {
		tmp = (((1.0 / (t_1 + Math.sqrt(y))) + 1.0) - Math.sqrt(x)) + t_3;
	} else {
		tmp = ((1.0 / (t_2 + Math.sqrt(z))) + (t_1 + 1.0)) - (Math.sqrt(y) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_4 = ((t_2 - math.sqrt(z)) + ((t_1 - math.sqrt(y)) + (math.sqrt((1.0 + x)) - math.sqrt(x)))) + t_3
	tmp = 0
	if t_4 <= 0.0:
		tmp = ((math.sqrt((1.0 / z)) + math.sqrt((1.0 / x))) * 0.5) + t_3
	elif t_4 <= 1.8:
		tmp = (((1.0 / (t_1 + math.sqrt(y))) + 1.0) - math.sqrt(x)) + t_3
	else:
		tmp = ((1.0 / (t_2 + math.sqrt(z))) + (t_1 + 1.0)) - (math.sqrt(y) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x)))) + t_3)
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / x))) * 0.5) + t_3);
	elseif (t_4 <= 1.8)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(t_1 + sqrt(y))) + 1.0) - sqrt(x)) + t_3);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_2 + sqrt(z))) + Float64(t_1 + 1.0)) - Float64(sqrt(y) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	t_4 = ((t_2 - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)))) + t_3;
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = ((sqrt((1.0 / z)) + sqrt((1.0 / x))) * 0.5) + t_3;
	elseif (t_4 <= 1.8)
		tmp = (((1.0 / (t_1 + sqrt(y))) + 1.0) - sqrt(x)) + t_3;
	else
		tmp = ((1.0 / (t_2 + sqrt(z))) + (t_1 + 1.0)) - (sqrt(y) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.8], N[(N[(N[(N[(1.0 / N[(t$95$1 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $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))) < 0.0

   1. Initial program 3.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 54.8%

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

   if 0.0 < (+.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))) < 1.80000000000000004

   1. Initial program 95.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 51.0

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

   7. Applied rewrites 51.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if 1.80000000000000004 < (+.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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 23.0%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \leq 1.8:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + 1\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ t_4 := \sqrt{y + 1}\\ t_5 := t\_1 - \sqrt{x}\\ t_6 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + t\_5\right)\\ t_7 := \frac{1}{t\_4 + \sqrt{y}}\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;t\_3 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, t\_7\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(\left(t\_7 + t\_1\right) - \sqrt{x}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{y}\right) + t\_5\right) + \frac{\left(z + 1\right) - z}{t\_2 + \sqrt{z}}\right) + t\_3\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (- t_1 (sqrt x)))
        (t_6 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt y)) t_5)))
        (t_7 (/ 1.0 (+ t_4 (sqrt y)))))
   (if (<= t_6 0.0)
     (+ t_3 (fma (sqrt (/ 1.0 x)) 0.5 t_7))
     (if (<= t_6 2.0)
       (+ (- (+ t_7 t_1) (sqrt x)) t_3)
       (+
        (+ (+ (- 1.0 (sqrt y)) t_5) (/ (- (+ z 1.0) z) (+ t_2 (sqrt z))))
        t_3)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double t_4 = sqrt((y + 1.0));
	double t_5 = t_1 - sqrt(x);
	double t_6 = (t_2 - sqrt(z)) + ((t_4 - sqrt(y)) + t_5);
	double t_7 = 1.0 / (t_4 + sqrt(y));
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_3 + fma(sqrt((1.0 / x)), 0.5, t_7);
	} else if (t_6 <= 2.0) {
		tmp = ((t_7 + t_1) - sqrt(x)) + t_3;
	} else {
		tmp = (((1.0 - sqrt(y)) + t_5) + (((z + 1.0) - z) / (t_2 + sqrt(z)))) + t_3;
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(1.0 / N[(t$95$4 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 0.0], N[(t$95$3 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$7), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(N[(t$95$7 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$2 + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.1%

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

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 39.4

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

   7. Applied rewrites 39.4%

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

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

   1. Initial program 94.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 91.7

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

   7. Applied rewrites 91.7%

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

4. Final simplification 49.3%

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

Alternative 8: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{t + 1}\\ t_3 := t\_2 - \sqrt{t}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_4 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\\ t_6 := \left(t\_1 - \sqrt{z}\right) + t\_5\\ \mathbf{if}\;t\_6 \leq 0:\\ \;\;\;\;t\_3 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{t\_4 + \sqrt{y}}\right)\\ \mathbf{elif}\;t\_6 \leq 2.0002:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} \cdot 0.5 + t\_5\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + t\_2} + \left(t\_1 + 1\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_2 (sqrt (+ t 1.0)))
        (t_3 (- t_2 (sqrt t)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_4 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))
        (t_6 (+ (- t_1 (sqrt z)) t_5)))
   (if (<= t_6 0.0)
     (+ t_3 (fma (sqrt (/ 1.0 x)) 0.5 (/ 1.0 (+ t_4 (sqrt y)))))
     (if (<= t_6 2.0002)
       (+ (+ (* (sqrt (/ 1.0 z)) 0.5) t_5) t_3)
       (+
        (-
         (+ (/ 1.0 (+ (sqrt t) t_2)) (+ t_1 1.0))
         (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
        1.0)))))

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));
	double t_2 = sqrt((t + 1.0));
	double t_3 = t_2 - sqrt(t);
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_4 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x));
	double t_6 = (t_1 - sqrt(z)) + t_5;
	double tmp;
	if (t_6 <= 0.0) {
		tmp = t_3 + fma(sqrt((1.0 / x)), 0.5, (1.0 / (t_4 + sqrt(y))));
	} else if (t_6 <= 2.0002) {
		tmp = ((sqrt((1.0 / z)) * 0.5) + t_5) + t_3;
	} else {
		tmp = (((1.0 / (sqrt(t) + t_2)) + (t_1 + 1.0)) - ((sqrt(z) + sqrt(y)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 46.8

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-sqrt.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.1%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00019999999999998

   1. Initial program 95.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 48.7

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

   5. Applied rewrites 48.7%

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

   if 2.00019999999999998 < (+.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 98.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 95.4%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 92.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.0002:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} \cdot 0.5 + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{t} + \sqrt{t + 1}} + \left(\sqrt{z + 1} + 1\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_2 - \sqrt{z}\right) + \left(\left(t\_4 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right)\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\left(t\_3 + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_5 \leq 2.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3 + x, 0.5, t\_4\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + 1\right) + t\_1\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ z 1.0)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (+ (- t_2 (sqrt z)) (+ (- t_4 (sqrt y)) (- t_1 (sqrt x))))))
   (if (<= t_5 0.0)
     (+ (* (+ t_3 (sqrt (/ 1.0 x))) 0.5) (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_5 2.0002)
       (+ (- (fma (+ t_3 x) 0.5 t_4) (+ (sqrt y) (sqrt x))) 1.0)
       (- (+ (+ t_2 1.0) t_1) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_2 - sqrt(z)) + ((t_4 - sqrt(y)) + (t_1 - sqrt(x)));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = ((t_3 + sqrt((1.0 / x))) * 0.5) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_5 <= 2.0002) {
		tmp = (fma((t_3 + x), 0.5, t_4) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = ((t_2 + 1.0) + t_1) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + x))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_2 - sqrt(z)) + Float64(Float64(t_4 - sqrt(y)) + Float64(t_1 - sqrt(x))))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(t_3 + sqrt(Float64(1.0 / x))) * 0.5) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_5 <= 2.0002)
		tmp = Float64(Float64(fma(Float64(t_3 + x), 0.5, t_4) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_2 + 1.0) + t_1) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[(N[(t$95$3 + N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0002], N[(N[(N[(N[(t$95$3 + x), $MachinePrecision] * 0.5 + t$95$4), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$2 + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 75.6%

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

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.00019999999999998

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate--r+ N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 23.2%

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

   if 2.00019999999999998 < (+.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 98.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.6%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 2.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}} + x, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} + 1\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((t + 1.0d0)) - sqrt(t)
    t_3 = sqrt((1.0d0 + x)) - sqrt(x)
    t_4 = (t_1 - sqrt(y)) + t_3
    if (t_4 <= 0.0d0) then
        tmp = ((sqrt((1.0d0 / z)) + sqrt((1.0d0 / x))) * 0.5d0) + t_2
    else if (t_4 <= 1.0d0) then
        tmp = t_3 + t_2
    else
        tmp = ((1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z))) + (t_1 + 1.0d0)) - (sqrt(y) + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double t_3 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double t_4 = (t_1 - Math.sqrt(y)) + t_3;
	double tmp;
	if (t_4 <= 0.0) {
		tmp = ((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / x))) * 0.5) + t_2;
	} else if (t_4 <= 1.0) {
		tmp = t_3 + t_2;
	} else {
		tmp = ((1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z))) + (t_1 + 1.0)) - (Math.sqrt(y) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((t + 1.0)) - math.sqrt(t)
	t_3 = math.sqrt((1.0 + x)) - math.sqrt(x)
	t_4 = (t_1 - math.sqrt(y)) + t_3
	tmp = 0
	if t_4 <= 0.0:
		tmp = ((math.sqrt((1.0 / z)) + math.sqrt((1.0 / x))) * 0.5) + t_2
	elif t_4 <= 1.0:
		tmp = t_3 + t_2
	else:
		tmp = ((1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z))) + (t_1 + 1.0)) - (math.sqrt(y) + math.sqrt(x))
	return tmp

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((t + 1.0)) - sqrt(t);
	t_3 = sqrt((1.0 + x)) - sqrt(x);
	t_4 = (t_1 - sqrt(y)) + t_3;
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = ((sqrt((1.0 / z)) + sqrt((1.0 / x))) * 0.5) + t_2;
	elseif (t_4 <= 1.0)
		tmp = t_3 + t_2;
	else
		tmp = ((1.0 / (sqrt((z + 1.0)) + sqrt(z))) + (t_1 + 1.0)) - (sqrt(y) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (+.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))) < 0.0

   1. Initial program 74.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \left(\sqrt{1 + x} + \sqrt{1 + y}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + x} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y} + \sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y} + \sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{y + 1}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{y + 1}} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lower-+.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{y + 1} + \sqrt{x + 1}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lower-sqrt.f64 27.6

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1} + \sqrt{x + 1}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1} + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 26.7%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 40.6%

       \[\leadsto 0.5 \cdot \left(\sqrt{\frac{1}{x}} + \color{blue}{\sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.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))) < 1

   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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{\color{blue}{x + 1}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{\color{blue}{x + 1}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 31.6

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{x + 1}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + x} - \sqrt{\color{blue}{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.5%

       \[\leadsto \left(\sqrt{x + 1} - \sqrt{\color{blue}{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.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)))

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   7. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 55.7%

       \[\leadsto \left(\left(1 + \sqrt{y + 1}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right) \leq 0:\\ \;\;\;\;\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{x}}\right) \cdot 0.5 + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right) \leq 1:\\ \;\;\;\;\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{y + 1} + 1\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_3 + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{t\_2 + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}} + \left(\left(t\_2 - \sqrt{y}\right) + t\_1\right)\right) + t\_3\\ \end{array} \end{array} \]

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)) (sqrt x)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_1 0.0)
     (+ t_3 (fma (sqrt (/ 1.0 x)) 0.5 (/ 1.0 (+ t_2 (sqrt y)))))
     (+
      (+
       (/ (- (+ z 1.0) z) (+ (sqrt (+ z 1.0)) (sqrt z)))
       (+ (- t_2 (sqrt y)) t_1))
      t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + x)) - sqrt(x);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_3 + fma(sqrt((1.0 / x)), 0.5, (1.0 / (t_2 + sqrt(y))));
	} else {
		tmp = ((((z + 1.0) - z) / (sqrt((z + 1.0)) + sqrt(z))) + ((t_2 - sqrt(y)) + t_1)) + t_3;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(t_3 + fma(sqrt(Float64(1.0 / x)), 0.5, Float64(1.0 / Float64(t_2 + sqrt(y)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(z + 1.0) - z) / Float64(sqrt(Float64(z + 1.0)) + sqrt(z))) + Float64(Float64(t_2 - sqrt(y)) + t_1)) + t_3);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(t$95$3 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$2 + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 87.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 88.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 88.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \sqrt{1 + x}\right)} - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \sqrt{1 + x}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\sqrt{1 + x}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. +-commutative N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{\color{blue}{x + 1}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{\color{blue}{x + 1}}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 19.8

          \[\leadsto \left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{x + 1}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 19.8%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \sqrt{x + 1}\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 47.6%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \color{blue}{0.5}, \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 96.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 96.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}} + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(t\_1 - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot x - \sqrt{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, t\_1\right) + 1\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))))
   (if (<=
        (+
         (- (sqrt (+ z 1.0)) (sqrt z))
         (+ (- t_1 (sqrt y)) (- (sqrt (+ 1.0 x)) (sqrt x))))
        1.0)
     (+ (- (* 0.5 x) (sqrt x)) 1.0)
     (- (+ (fma 0.5 x t_1) 1.0) (+ (sqrt y) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double tmp;
	if (((sqrt((z + 1.0)) - sqrt(z)) + ((t_1 - sqrt(y)) + (sqrt((1.0 + x)) - sqrt(x)))) <= 1.0) {
		tmp = ((0.5 * x) - sqrt(x)) + 1.0;
	} else {
		tmp = (fma(0.5, x, t_1) + 1.0) - (sqrt(y) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(z)) + Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(1.0 + x)) - sqrt(x)))) <= 1.0)
		tmp = Float64(Float64(Float64(0.5 * x) - sqrt(x)) + 1.0);
	else
		tmp = Float64(Float64(fma(0.5, x, t_1) + 1.0) - Float64(sqrt(y) + sqrt(x)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[(N[(0.5 * x), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * x + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $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

   1. Initial program 86.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.0%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto 1 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.4%

       \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) \]

   12. Taylor expanded in z around inf

       \[\leadsto 1 + \left(\frac{1}{2} \cdot x - \sqrt{x}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 21.2%

       \[\leadsto 1 + \left(0.5 \cdot x - \sqrt{x}\right) \]

   if 1 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 15.5%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.0%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto \left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + 1\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{z + 1} - \sqrt{z}\right) + \left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot x - \sqrt{x}\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) + 1\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5800000:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}} + x, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 5800000.0)
   (- (+ (fma 0.5 x (sqrt (+ z 1.0))) 2.0) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
   (+
    (- (fma (+ (sqrt (/ 1.0 z)) x) 0.5 (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x)))
    1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5800000.0) {
		tmp = (fma(0.5, x, sqrt((z + 1.0))) + 2.0) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	} else {
		tmp = (fma((sqrt((1.0 / z)) + x), 0.5, sqrt((y + 1.0))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5800000.0)
		tmp = Float64(Float64(fma(0.5, x, sqrt(Float64(z + 1.0))) + 2.0) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = Float64(Float64(fma(Float64(sqrt(Float64(1.0 / z)) + x), 0.5, sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	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[z, 5800000.0], N[(N[(N[(0.5 * x + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision] * 0.5 + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e6

   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 x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.6%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 16.0%

       \[\leadsto \left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   if 5.8e6 < z

   1. Initial program 86.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \left(\frac{1}{2} \cdot x + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 36.7%

       \[\leadsto 1 + \left(\mathsf{fma}\left(x + \sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5800000:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}} + x, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 3.55 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.55e+15)
   (- (+ (fma 0.5 x (sqrt (+ z 1.0))) 2.0) (+ (+ (sqrt z) (sqrt y)) (sqrt x)))
   (+ (- (fma 0.5 x (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x))) 1.0)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.55e+15) {
		tmp = (fma(0.5, x, sqrt((z + 1.0))) + 2.0) - ((sqrt(z) + sqrt(y)) + sqrt(x));
	} else {
		tmp = (fma(0.5, x, sqrt((y + 1.0))) - (sqrt(y) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.55e+15)
		tmp = Float64(Float64(fma(0.5, x, sqrt(Float64(z + 1.0))) + 2.0) - Float64(Float64(sqrt(z) + sqrt(y)) + sqrt(x)));
	else
		tmp = Float64(Float64(fma(0.5, x, sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	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[z, 3.55e+15], N[(N[(N[(0.5 * x + N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.55e15

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 16.0%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.8%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 16.1%

       \[\leadsto \left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   if 3.55e15 < z

   1. Initial program 86.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

      3. associate-+r+ N/A

         \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

      4. associate--r+ N/A

         \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

      5. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.8%

      \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 36.1%

       \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.55 \cdot 10^{+15}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (fma 0.5 x (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x))) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (fma(0.5, x, sqrt((y + 1.0))) - (sqrt(y) + sqrt(x))) + 1.0;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(fma(0.5, x, sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x))) + 1.0)
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

   3. associate-+r+ N/A

      \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

   4. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

   5. associate-+l- N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

5. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 25.6%

   \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

9. Taylor expanded in z around inf

   \[\leadsto 1 + \left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 23.6%

    \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{x} + \color{blue}{\sqrt{y}}\right)\right) \]

12. Final simplification 23.6%

    \[\leadsto \left(\mathsf{fma}\left(0.5, x, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1 \]
13. Add Preprocessing

Alternative 16: 35.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(0.5 \cdot x - \sqrt{x}\right) + 1 \end{array} \]

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) (sqrt x)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return ((0.5 * x) - sqrt(x)) + 1.0;
}

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) - sqrt(x)) + 1.0d0
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) - Math.sqrt(x)) + 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return ((0.5 * x) - math.sqrt(x)) + 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - sqrt(x)) + 1.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = ((0.5 * x) - sqrt(x)) + 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) + 1} \]

   3. associate-+r+ N/A

      \[\leadsto \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\left(\sqrt{t} + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) + 1 \]

   4. associate--r+ N/A

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\right)} + 1 \]

   5. associate-+l- N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{t} + \sqrt{x}\right)\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) - 1\right)} \]

5. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) + \sqrt{t + 1}\right) - \left(\sqrt{x} + \sqrt{t}\right)\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) - 1\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 25.6%

   \[\leadsto 1 + \color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto 1 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right) - \left(\sqrt{x} + \color{blue}{\sqrt{z}}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 25.5%

    \[\leadsto 1 + \left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) \]

12. Taylor expanded in z around inf

    \[\leadsto 1 + \left(\frac{1}{2} \cdot x - \sqrt{x}\right) \]
13. Step-by-step derivation

14. Applied rewrites 17.4%

    \[\leadsto 1 + \left(0.5 \cdot x - \sqrt{x}\right) \]

15. Final simplification 17.4%

    \[\leadsto \left(0.5 \cdot x - \sqrt{x}\right) + 1 \]
16. Add Preprocessing

Alternative 17: 34.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(-\sqrt{x}\right) + 1 \end{array} \]

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 x)) 1.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x) + 1.0;
}

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(x) + 1.0d0
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(x) + 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x) + 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(-sqrt(x)) + 1.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x) + 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[((-N[Sqrt[x], $MachinePrecision]) + 1.0), $MachinePrecision]

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

   2. flip-- N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   6. rem-square-sqrt N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

   9. rem-square-sqrt N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   10. lower--.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

   11. +-commutative N/A

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   12. lower-+.f64 91.9

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

4. Applied rewrites 91.9%

   \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

7. Applied rewrites 32.9%

   \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto 1 + -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 15.3%

    \[\leadsto 1 + \left(-\sqrt{x}\right) \]

11. Final simplification 15.3%

    \[\leadsto \left(-\sqrt{x}\right) + 1 \]
12. Add Preprocessing

Alternative 18: 1.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -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 = -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 -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -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[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 91.8%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

   2. flip-- N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   6. rem-square-sqrt N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

   9. rem-square-sqrt N/A

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

   10. lower--.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

   11. +-commutative N/A

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   12. lower-+.f64 91.9

       \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

4. Applied rewrites 91.9%

   \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

7. Applied rewrites 32.9%

   \[\leadsto \color{blue}{1 + \left(\left(\left(\frac{1}{\sqrt{t + 1} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \color{blue}{\sqrt{x}} \]
9. Step-by-step derivation

10. Applied rewrites 1.6%

    \[\leadsto -\sqrt{x} \]
11. 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 2024243 
(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))))