Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 69.4% → 99.9%

Time: 7.3s

Alternatives: 10

Speedup: 1.2×

• 40.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \mathsf{fma}\left(\frac{z - x}{y}, z + x, -y\right) \cdot -0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z) :precision binary64 (* (fma (/ (- z x) y) (+ z x) (- y)) -0.5))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	return fma(((z - x) / y), (z + x), -y) * -0.5;
}

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	return Float64(fma(Float64(Float64(z - x) / y), Float64(z + x), Float64(-y)) * -0.5)
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := N[(N[(N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] * N[(z + x), $MachinePrecision] + (-y)), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

      \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

   4. associate-+l- 70.2%

      \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

   5. sub0-neg 70.2%

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

   6. neg-mul-1 70.2%

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

   7. *-commutative 70.2%

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

   8. times-frac 70.2%

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

   9. associate--r+ 70.2%

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

   10. div-sub 70.2%

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

   11. difference-of-squares 74.1%

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

   12. +-commutative 74.1%

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

   13. associate-*r/ 80.1%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
4. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. fma-neg 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+266}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{y} \cdot \left(z + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-73)
   (* -0.5 (- (- y) (* x (/ x y))))
   (if (<= (* z z) 5e+266)
     (* -0.5 (* (/ (- z x) y) (+ z x)))
     (* -0.5 (- (/ z (/ y z)) y)))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-73) {
		tmp = -0.5 * (-y - (x * (x / y)));
	} else if ((z * z) <= 5e+266) {
		tmp = -0.5 * (((z - x) / y) * (z + x));
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-73) then
        tmp = (-0.5d0) * (-y - (x * (x / y)))
    else if ((z * z) <= 5d+266) then
        tmp = (-0.5d0) * (((z - x) / y) * (z + x))
    else
        tmp = (-0.5d0) * ((z / (y / z)) - y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-73) {
		tmp = -0.5 * (-y - (x * (x / y)));
	} else if ((z * z) <= 5e+266) {
		tmp = -0.5 * (((z - x) / y) * (z + x));
	} else {
		tmp = -0.5 * ((z / (y / z)) - y);
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-73:
		tmp = -0.5 * (-y - (x * (x / y)))
	elif (z * z) <= 5e+266:
		tmp = -0.5 * (((z - x) / y) * (z + x))
	else:
		tmp = -0.5 * ((z / (y / z)) - y)
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-73)
		tmp = Float64(-0.5 * Float64(Float64(-y) - Float64(x * Float64(x / y))));
	elseif (Float64(z * z) <= 5e+266)
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) / y) * Float64(z + x)));
	else
		tmp = Float64(-0.5 * Float64(Float64(z / Float64(y / z)) - y));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-73)
		tmp = -0.5 * (-y - (x * (x / y)));
	elseif ((z * z) <= 5e+266)
		tmp = -0.5 * (((z - x) / y) * (z + x));
	else
		tmp = -0.5 * ((z / (y / z)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-73], N[(-0.5 * N[((-y) - N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+266], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.99999999999999999e-73

   1. Initial program 72.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 72.9%

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

      2. +-commutative 72.9%

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

      3. neg-sub0 72.9%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 72.9%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 72.9%

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

      6. neg-mul-1 72.9%

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

      7. *-commutative 72.9%

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

      8. times-frac 72.9%

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

      9. associate--r+ 72.9%

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

      10. div-sub 72.9%

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

      11. difference-of-squares 72.9%

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

      12. +-commutative 72.9%

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

      13. associate-*r/ 78.4%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 87.3%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   5. Step-by-step derivation

      1. unpow2 87.3%

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

      2. associate-*r/ 87.3%

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

      3. neg-mul-1 87.3%

         \[\leadsto \left(\frac{\color{blue}{-x \cdot x}}{y} - y\right) \cdot -0.5 \]

      4. distribute-rgt-neg-in 87.3%

         \[\leadsto \left(\frac{\color{blue}{x \cdot \left(-x\right)}}{y} - y\right) \cdot -0.5 \]

   6. Simplified 87.3%

      \[\leadsto \left(\color{blue}{\frac{x \cdot \left(-x\right)}{y}} - y\right) \cdot -0.5 \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 87.3%

         \[\leadsto \left(\frac{\color{blue}{-x \cdot x}}{y} - y\right) \cdot -0.5 \]

      2. distribute-frac-neg 87.3%

         \[\leadsto \left(\color{blue}{\left(-\frac{x \cdot x}{y}\right)} - y\right) \cdot -0.5 \]

      3. div-inv 87.3%

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

      4. add-sqr-sqrt 87.3%

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

      5. sqrt-unprod 78.6%

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

      6. sqr-neg 78.6%

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

      7. swap-sqr 78.6%

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

      8. sqrt-unprod 20.7%

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

      9. add-sqr-sqrt 42.2%

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

      10. associate-*l* 46.4%

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

      11. add-sqr-sqrt 25.7%

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

      12. sqrt-unprod 60.6%

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

      13. sqr-neg 60.6%

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

      14. sqrt-unprod 44.3%

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

      15. add-sqr-sqrt 98.2%

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

      16. div-inv 98.3%

          \[\leadsto \left(\left(-x \cdot \color{blue}{\frac{-x}{-y}}\right) - y\right) \cdot -0.5 \]

      17. frac-2neg 98.3%

          \[\leadsto \left(\left(-x \cdot \color{blue}{\frac{x}{y}}\right) - y\right) \cdot -0.5 \]

   8. Applied egg-rr 98.3%

      \[\leadsto \left(\color{blue}{\left(-x \cdot \frac{x}{y}\right)} - y\right) \cdot -0.5 \]

   if 1.99999999999999999e-73 < (*.f64 z z) < 4.9999999999999999e266

   1. Initial program 79.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 79.5%

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

      2. +-commutative 79.5%

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

      3. neg-sub0 79.5%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 79.5%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 79.5%

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

      6. neg-mul-1 79.5%

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

      7. *-commutative 79.5%

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

      8. times-frac 79.5%

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

      9. associate--r+ 79.5%

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

      10. div-sub 79.6%

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

      11. difference-of-squares 79.6%

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

      12. +-commutative 79.6%

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

      13. associate-*r/ 84.3%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 72.0%

      \[\leadsto \color{blue}{\frac{\left(z - x\right) \cdot \left(z + x\right)}{y}} \cdot -0.5 \]
   7. Step-by-step derivation

      1. associate-/l* 77.6%

         \[\leadsto \color{blue}{\frac{z - x}{\frac{y}{z + x}}} \cdot -0.5 \]

      2. associate-/r/ 77.7%

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

      3. +-commutative 77.7%

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

   8. Simplified 77.7%

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

   if 4.9999999999999999e266 < (*.f64 z z)

   1. Initial program 59.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 59.0%

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

      2. +-commutative 59.0%

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

      3. neg-sub0 59.0%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 59.0%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 59.0%

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

      6. neg-mul-1 59.0%

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

      7. *-commutative 59.0%

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

      8. times-frac 59.0%

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

      9. associate--r+ 59.0%

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

      10. div-sub 59.0%

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

      11. difference-of-squares 71.8%

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

      12. +-commutative 71.8%

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

      13. associate-*r/ 79.3%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{y} - y\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \left(\frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot -0.5 \]

      2. associate-/l* 91.1%

         \[\leadsto \left(\color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot -0.5 \]

   8. Simplified 91.1%

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{y}{z}} - y\right)} \cdot -0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - x \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+266}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{y} \cdot \left(z + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 3: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e-48)
   (* -0.5 (- (* z (/ z y)) y))
   (* -0.5 (- (* (- z x) (/ x y)) y))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e-48) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d-48) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (-0.5d0) * (((z - x) * (x / y)) - y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e-48) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e-48:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e-48)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e-48)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-48], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999999e-48

   1. Initial program 75.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 75.5%

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

      2. +-commutative 75.5%

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

      3. neg-sub0 75.5%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 75.5%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. times-frac 75.5%

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

      9. associate--r+ 75.5%

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

      10. div-sub 75.5%

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

      11. difference-of-squares 75.5%

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

      12. +-commutative 75.5%

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

      13. associate-*r/ 80.8%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{y} - y\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 84.7%

         \[\leadsto \left(\frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot -0.5 \]

      2. associate-/l* 91.3%

         \[\leadsto \left(\color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot -0.5 \]

   8. Simplified 91.3%

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{y}{z}} - y\right)} \cdot -0.5 \]
   9. Step-by-step derivation

      1. associate-/r/ 91.3%

         \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   10. Applied egg-rr 91.3%

       \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   if 1.9999999999999999e-48 < (*.f64 x x)

   1. Initial program 64.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 64.7%

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

      2. +-commutative 64.7%

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

      3. neg-sub0 64.7%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 64.7%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 64.7%

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

      6. neg-mul-1 64.7%

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

      7. *-commutative 64.7%

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

      8. times-frac 64.7%

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

      9. associate--r+ 64.7%

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

      10. div-sub 64.7%

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

      11. difference-of-squares 72.7%

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

      12. +-commutative 72.7%

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

      13. associate-*l/ 79.4%

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

      14. *-commutative 79.4%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 87.0%

      \[\leadsto \left(\left(z - x\right) \cdot \color{blue}{\frac{x}{y}} - y\right) \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - x \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 5e+117)
   (* -0.5 (- (* z (/ z y)) y))
   (* -0.5 (- (- y) (* x (/ x y))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+117) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (-y - (x * (x / y)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d+117) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (-0.5d0) * (-y - (x * (x / y)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 5e+117) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * (-y - (x * (x / y)));
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 5e+117:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = -0.5 * (-y - (x * (x / y)))
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+117)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(-y) - Float64(x * Float64(x / y))));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 5e+117)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = -0.5 * (-y - (x * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+117], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[((-y) - N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999983e117

   1. Initial program 75.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 75.0%

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

      2. +-commutative 75.0%

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

      3. neg-sub0 75.0%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 75.0%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 75.0%

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

      6. neg-mul-1 75.0%

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

      7. *-commutative 75.0%

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

      8. times-frac 75.0%

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

      9. associate--r+ 75.0%

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

      10. div-sub 75.0%

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

      11. difference-of-squares 75.0%

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

      12. +-commutative 75.0%

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

      13. associate-*r/ 79.5%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 83.8%

      \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{y} - y\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 83.8%

         \[\leadsto \left(\frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot -0.5 \]

      2. associate-/l* 89.4%

         \[\leadsto \left(\color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot -0.5 \]

   8. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{y}{z}} - y\right)} \cdot -0.5 \]
   9. Step-by-step derivation

      1. associate-/r/ 89.4%

         \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   10. Applied egg-rr 89.4%

       \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   if 4.99999999999999983e117 < (*.f64 x x)

   1. Initial program 62.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 62.9%

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

      2. +-commutative 62.9%

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

      3. neg-sub0 62.9%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 62.9%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 62.9%

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

      6. neg-mul-1 62.9%

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

      7. *-commutative 62.9%

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

      8. times-frac 62.9%

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

      9. associate--r+ 62.9%

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

      10. div-sub 62.9%

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

      11. difference-of-squares 72.8%

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

      12. +-commutative 72.8%

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

      13. associate-*r/ 81.0%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 75.3%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   5. Step-by-step derivation

      1. unpow2 75.4%

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

      2. associate-*r/ 75.4%

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

      3. neg-mul-1 75.4%

         \[\leadsto \left(\frac{\color{blue}{-x \cdot x}}{y} - y\right) \cdot -0.5 \]

      4. distribute-rgt-neg-in 75.4%

         \[\leadsto \left(\frac{\color{blue}{x \cdot \left(-x\right)}}{y} - y\right) \cdot -0.5 \]

   6. Simplified 75.4%

      \[\leadsto \left(\color{blue}{\frac{x \cdot \left(-x\right)}{y}} - y\right) \cdot -0.5 \]
   7. Step-by-step derivation

      1. distribute-rgt-neg-out 75.4%

         \[\leadsto \left(\frac{\color{blue}{-x \cdot x}}{y} - y\right) \cdot -0.5 \]

      2. distribute-frac-neg 75.4%

         \[\leadsto \left(\color{blue}{\left(-\frac{x \cdot x}{y}\right)} - y\right) \cdot -0.5 \]

      3. div-inv 75.4%

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

      4. add-sqr-sqrt 75.4%

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

      5. sqrt-unprod 60.8%

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

      6. sqr-neg 60.8%

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

      7. swap-sqr 60.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 15.8%

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

      10. associate-*l* 20.7%

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

      11. add-sqr-sqrt 10.8%

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

      12. sqrt-unprod 51.9%

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

      13. sqr-neg 51.9%

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

      14. sqrt-unprod 44.1%

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

      15. add-sqr-sqrt 89.7%

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

      16. div-inv 89.8%

          \[\leadsto \left(\left(-x \cdot \color{blue}{\frac{-x}{-y}}\right) - y\right) \cdot -0.5 \]

      17. frac-2neg 89.8%

          \[\leadsto \left(\left(-x \cdot \color{blue}{\frac{x}{y}}\right) - y\right) \cdot -0.5 \]

   8. Applied egg-rr 89.8%

      \[\leadsto \left(\color{blue}{\left(-x \cdot \frac{x}{y}\right)} - y\right) \cdot -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 5: 52.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-233}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 17500000:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 3.5e-233)
   (* y 0.5)
   (if (<= x 17500000.0)
     (* -0.5 (/ z (/ y z)))
     (if (<= x 2.7e+62) (* y 0.5) (* (/ x y) (* x 0.5))))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.5e-233) {
		tmp = y * 0.5;
	} else if (x <= 17500000.0) {
		tmp = -0.5 * (z / (y / z));
	} else if (x <= 2.7e+62) {
		tmp = y * 0.5;
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.5d-233) then
        tmp = y * 0.5d0
    else if (x <= 17500000.0d0) then
        tmp = (-0.5d0) * (z / (y / z))
    else if (x <= 2.7d+62) then
        tmp = y * 0.5d0
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.5e-233) {
		tmp = y * 0.5;
	} else if (x <= 17500000.0) {
		tmp = -0.5 * (z / (y / z));
	} else if (x <= 2.7e+62) {
		tmp = y * 0.5;
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 3.5e-233:
		tmp = y * 0.5
	elif x <= 17500000.0:
		tmp = -0.5 * (z / (y / z))
	elif x <= 2.7e+62:
		tmp = y * 0.5
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 3.5e-233)
		tmp = Float64(y * 0.5);
	elseif (x <= 17500000.0)
		tmp = Float64(-0.5 * Float64(z / Float64(y / z)));
	elseif (x <= 2.7e+62)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.5e-233)
		tmp = y * 0.5;
	elseif (x <= 17500000.0)
		tmp = -0.5 * (z / (y / z));
	elseif (x <= 2.7e+62)
		tmp = y * 0.5;
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 3.5e-233], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 17500000.0], N[(-0.5 * N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+62], N[(y * 0.5), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.49999999999999991e-233 or 1.75e7 < x < 2.7e62

   1. Initial program 73.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf 33.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 33.4%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   4. Simplified 33.4%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 3.49999999999999991e-233 < x < 1.75e7

   1. Initial program 74.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 74.9%

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

      2. +-commutative 74.9%

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

      3. neg-sub0 74.9%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 74.9%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 74.9%

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

      6. neg-mul-1 74.9%

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

      7. *-commutative 74.9%

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

      8. times-frac 74.9%

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

      9. associate--r+ 74.9%

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

      10. div-sub 74.9%

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

      11. difference-of-squares 74.9%

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

      12. +-commutative 74.9%

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

      13. associate-*r/ 83.2%

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

      14. associate-/l* 99.8%

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

      15. *-inverses 99.8%

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

      16. /-rgt-identity 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. fma-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around inf 45.4%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y}} \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 45.4%

         \[\leadsto \frac{\color{blue}{z \cdot z}}{y} \cdot -0.5 \]

      2. associate-/l* 52.3%

         \[\leadsto \color{blue}{\frac{z}{\frac{y}{z}}} \cdot -0.5 \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{z}{\frac{y}{z}}} \cdot -0.5 \]

   if 2.7e62 < x

   1. Initial program 56.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf 56.3%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]

   4. Simplified 56.3%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. Step-by-step derivation

      1. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{2}} \]

      2. div-inv 68.3%

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

      3. metadata-eval 68.3%

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

   6. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-233}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 17500000:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+192}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+192) (* -0.5 (- (* z (/ z y)) y)) (* (/ x y) (* x 0.5))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+192) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 1d+192) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+192) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+192:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+192)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+192)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+192], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000004e192

   1. Initial program 73.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. sub-neg 73.6%

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

      2. +-commutative 73.6%

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

      3. neg-sub0 73.6%

         \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

      4. associate-+l- 73.6%

         \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

      5. sub0-neg 73.6%

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

      6. neg-mul-1 73.6%

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

      7. *-commutative 73.6%

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

      8. times-frac 73.6%

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

      9. associate--r+ 73.6%

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

      10. div-sub 73.7%

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

      11. difference-of-squares 73.7%

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

      12. +-commutative 73.7%

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

      13. associate-*r/ 77.9%

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

      14. associate-/l* 99.9%

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

      15. *-inverses 99.9%

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

      16. /-rgt-identity 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right) \cdot -0.5} \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{y} - y\right)} \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto \left(\frac{\color{blue}{z \cdot z}}{y} - y\right) \cdot -0.5 \]

      2. associate-/l* 87.6%

         \[\leadsto \left(\color{blue}{\frac{z}{\frac{y}{z}}} - y\right) \cdot -0.5 \]

   8. Simplified 87.6%

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{y}{z}} - y\right)} \cdot -0.5 \]
   9. Step-by-step derivation

      1. associate-/r/ 87.6%

         \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   10. Applied egg-rr 87.6%

       \[\leadsto \left(\color{blue}{\frac{z}{y} \cdot z} - y\right) \cdot -0.5 \]

   if 1.00000000000000004e192 < (*.f64 x x)

   1. Initial program 64.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf 68.4%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. unpow2 68.4%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]

   4. Simplified 68.4%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. Step-by-step derivation

      1. times-frac 77.9%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{2}} \]

      2. div-inv 77.9%

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

      3. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+192}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ -0.5 \cdot \left(\frac{z - x}{y} \cdot \left(z + x\right) - y\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z) :precision binary64 (* -0.5 (- (* (/ (- z x) y) (+ z x)) y)))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	return -0.5 * ((((z - x) / y) * (z + x)) - y);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.5d0) * ((((z - x) / y) * (z + x)) - y)
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	return -0.5 * ((((z - x) / y) * (z + x)) - y);
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	return -0.5 * ((((z - x) / y) * (z + x)) - y)

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	return Float64(-0.5 * Float64(Float64(Float64(Float64(z - x) / y) * Float64(z + x)) - y))
end

MATLAB

x = abs(x)
z = abs(z)
function tmp = code(x, y, z)
	tmp = -0.5 * ((((z - x) / y) * (z + x)) - y);
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := N[(-0.5 * N[(N[(N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. sub-neg 70.2%

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

   2. +-commutative 70.2%

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

   3. neg-sub0 70.2%

      \[\leadsto \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]

   4. associate-+l- 70.2%

      \[\leadsto \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]

   5. sub0-neg 70.2%

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

   6. neg-mul-1 70.2%

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

   7. *-commutative 70.2%

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

   8. times-frac 70.2%

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

   9. associate--r+ 70.2%

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

   10. div-sub 70.2%

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

   11. difference-of-squares 74.1%

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

   12. +-commutative 74.1%

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

   13. associate-*r/ 80.1%

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

   14. associate-/l* 99.9%

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

   15. *-inverses 99.9%

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

   16. /-rgt-identity 99.9%

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

   17. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto -0.5 \cdot \left(\frac{z - x}{y} \cdot \left(z + x\right) - y\right) \]

Alternative 8: 52.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 2.8e+62) (* y 0.5) (* x (* x (/ 0.5 y)))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+62) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.8d+62) then
        tmp = y * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.8e+62) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 2.8e+62:
		tmp = y * 0.5
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 2.8e+62)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.8e+62)
		tmp = y * 0.5;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 2.8e+62], N[(y * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000014e62

   1. Initial program 73.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf 34.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   4. Simplified 34.3%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 2.80000000000000014e62 < x

   1. Initial program 56.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf 56.3%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]

   4. Simplified 56.3%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]

   5. Taylor expanded in x around 0 56.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 56.3%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. unpow2 56.3%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \cdot 0.5 \]

      3. metadata-eval 56.3%

         \[\leadsto \frac{x \cdot x}{y} \cdot \color{blue}{\frac{1}{2}} \]

      4. times-frac 56.3%

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

      5. associate-*r/ 56.3%

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

      6. associate-*r* 68.2%

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

      7. *-commutative 68.2%

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

      8. associate-/r* 68.2%

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

      9. metadata-eval 68.2%

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

   7. Simplified 68.2%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 9: 52.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 2.9e+62) (* y 0.5) (* (/ x y) (* x 0.5))))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.9e+62) {
		tmp = y * 0.5;
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.9d+62) then
        tmp = y * 0.5d0
    else
        tmp = (x / y) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.9e+62) {
		tmp = y * 0.5;
	} else {
		tmp = (x / y) * (x * 0.5);
	}
	return tmp;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	tmp = 0
	if x <= 2.9e+62:
		tmp = y * 0.5
	else:
		tmp = (x / y) * (x * 0.5)
	return tmp

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	tmp = 0.0
	if (x <= 2.9e+62)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(x / y) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

x = abs(x)
z = abs(z)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.9e+62)
		tmp = y * 0.5;
	else
		tmp = (x / y) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 2.9e+62], N[(y * 0.5), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999984e62

   1. Initial program 73.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf 34.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   4. Simplified 34.3%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 2.89999999999999984e62 < x

   1. Initial program 56.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in x around inf 56.3%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot 2} \]
   3. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]

   4. Simplified 56.3%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. Step-by-step derivation

      1. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{2}} \]

      2. div-inv 68.3%

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

      3. metadata-eval 68.3%

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

   6. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 35.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} x = |x|\\ z = |z|\\ \\ y \cdot 0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
(FPCore (x y z) :precision binary64 (* y 0.5))

C

x = abs(x);
z = abs(z);
double code(double x, double y, double z) {
	return y * 0.5;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * 0.5d0
end function

Java

x = Math.abs(x);
z = Math.abs(z);
public static double code(double x, double y, double z) {
	return y * 0.5;
}

Python

x = abs(x)
z = abs(z)
def code(x, y, z):
	return y * 0.5

Julia

x = abs(x)
z = abs(z)
function code(x, y, z)
	return Float64(y * 0.5)
end

MATLAB

x = abs(x)
z = abs(z)
function tmp = code(x, y, z)
	tmp = y * 0.5;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: z should be positive before calling this function
code[x_, y_, z_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

2. Taylor expanded in y around inf 32.0%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Step-by-step derivation

   1. *-commutative 32.0%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

4. Simplified 32.0%

   \[\leadsto \color{blue}{y \cdot 0.5} \]

5. Final simplification 32.0%

   \[\leadsto y \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))