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

Percentage Accurate: 68.7% → 84.1%

Time: 10.3s

Alternatives: 10

Speedup: 0.7×

• 41.4% 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: 68.7% 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: 84.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;z \cdot z \leq 10^{+304}:\\ \;\;\;\;0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \frac{x}{{z}^{2}}, \frac{-1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, \frac{1}{z} \cdot \frac{x}{z}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-136)
   (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
   (if (<= (* z z) 1e+304)
     (*
      0.5
      (*
       (pow z 2.0)
       (+ (/ y (pow z 2.0)) (fma (/ x y) (/ x (pow z 2.0)) (/ -1.0 y)))))
     (* 0.5 (* (fma x (* (/ 1.0 z) (/ x z)) -1.0) (* z (* z (/ 1.0 y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-136) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if ((z * z) <= 1e+304) {
		tmp = 0.5 * (pow(z, 2.0) * ((y / pow(z, 2.0)) + fma((x / y), (x / pow(z, 2.0)), (-1.0 / y))));
	} else {
		tmp = 0.5 * (fma(x, ((1.0 / z) * (x / z)), -1.0) * (z * (z * (1.0 / y))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-136)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (Float64(z * z) <= 1e+304)
		tmp = Float64(0.5 * Float64((z ^ 2.0) * Float64(Float64(y / (z ^ 2.0)) + fma(Float64(x / y), Float64(x / (z ^ 2.0)), Float64(-1.0 / y)))));
	else
		tmp = Float64(0.5 * Float64(fma(x, Float64(Float64(1.0 / z) * Float64(x / z)), -1.0) * Float64(z * Float64(z * Float64(1.0 / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-136], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+304], N[(0.5 * N[(N[Power[z, 2.0], $MachinePrecision] * N[(N[(y / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(x / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x * N[(N[(1.0 / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(z * N[(z * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e-136

   1. Initial program 79.2%

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

   if 2e-136 < (*.f64 z z) < 9.9999999999999994e303

   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. remove-double-neg 75.0%

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

      2. distribute-lft-neg-out 75.0%

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

      3. distribute-frac-neg2 75.0%

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

      4. distribute-frac-neg 75.0%

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

      5. neg-mul-1 75.0%

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

      6. distribute-lft-neg-out 75.0%

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

      7. *-commutative 75.0%

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

      8. distribute-lft-neg-in 75.0%

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

      9. times-frac 75.0%

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

      10. metadata-eval 75.0%

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

      11. metadata-eval 75.0%

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

      12. associate--l+ 75.0%

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

      13. fma-define 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in z around inf 87.8%

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

      1. associate--l+ 87.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 87.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 98.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   if 9.9999999999999994e303 < (*.f64 z z)

   1. Initial program 57.2%

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

      1. remove-double-neg 57.2%

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

      2. distribute-lft-neg-out 57.2%

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

      3. distribute-frac-neg2 57.2%

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

      4. distribute-frac-neg 57.2%

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

      5. neg-mul-1 57.2%

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

      6. distribute-lft-neg-out 57.2%

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

      7. *-commutative 57.2%

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

      8. distribute-lft-neg-in 57.2%

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

      9. times-frac 57.2%

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

      10. metadata-eval 57.2%

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

      11. metadata-eval 57.2%

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

      12. associate--l+ 57.2%

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

      13. fma-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in z around inf 65.2%

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

      1. associate--l+ 65.2%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 65.2%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 66.7%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 66.7%

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

      5. distribute-neg-frac 66.7%

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

      6. metadata-eval 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in y around 0 65.2%

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

      1. *-commutative 65.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot {z}^{2}}}{y} \]

      2. associate-/l* 65.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right)} \]

      3. unpow2 65.2%

         \[\leadsto 0.5 \cdot \left(\left(\frac{\color{blue}{x \cdot x}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      4. associate-/l* 69.8%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{x \cdot \frac{x}{{z}^{2}}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      5. fmm-def 69.8%

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

      6. metadata-eval 69.8%

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

   10. Simplified 69.8%

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

       1. div-inv 69.8%

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

       2. pow2 69.8%

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

       3. associate-*l* 74.2%

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

   12. Applied egg-rr 74.2%

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

       1. *-un-lft-identity 74.2%

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

       2. pow2 74.2%

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

       3. times-frac 89.5%

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

   14. Applied egg-rr 89.5%

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(x, \color{blue}{\frac{1}{z} \cdot \frac{x}{z}}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{z} \cdot \frac{x}{z}\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;z \cdot z \leq 10^{+304}:\\ \;\;\;\;0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, t\_0, \frac{-1}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, t\_0, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 z) (/ x z))))
   (if (<= (* z z) 2e-136)
     (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
     (if (<= (* z z) 1e+304)
       (*
        0.5
        (* (pow z 2.0) (+ (/ y (pow z 2.0)) (fma (/ x y) t_0 (/ -1.0 y)))))
       (* 0.5 (* (fma x t_0 -1.0) (* z (* z (/ 1.0 y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 / z) * (x / z);
	double tmp;
	if ((z * z) <= 2e-136) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if ((z * z) <= 1e+304) {
		tmp = 0.5 * (pow(z, 2.0) * ((y / pow(z, 2.0)) + fma((x / y), t_0, (-1.0 / y))));
	} else {
		tmp = 0.5 * (fma(x, t_0, -1.0) * (z * (z * (1.0 / y))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 / z) * Float64(x / z))
	tmp = 0.0
	if (Float64(z * z) <= 2e-136)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (Float64(z * z) <= 1e+304)
		tmp = Float64(0.5 * Float64((z ^ 2.0) * Float64(Float64(y / (z ^ 2.0)) + fma(Float64(x / y), t_0, Float64(-1.0 / y)))));
	else
		tmp = Float64(0.5 * Float64(fma(x, t_0, -1.0) * Float64(z * Float64(z * Float64(1.0 / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e-136], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+304], N[(0.5 * N[(N[Power[z, 2.0], $MachinePrecision] * N[(N[(y / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * t$95$0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x * t$95$0 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e-136

   1. Initial program 79.2%

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

   if 2e-136 < (*.f64 z z) < 9.9999999999999994e303

   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. remove-double-neg 75.0%

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

      2. distribute-lft-neg-out 75.0%

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

      3. distribute-frac-neg2 75.0%

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

      4. distribute-frac-neg 75.0%

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

      5. neg-mul-1 75.0%

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

      6. distribute-lft-neg-out 75.0%

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

      7. *-commutative 75.0%

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

      8. distribute-lft-neg-in 75.0%

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

      9. times-frac 75.0%

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

      10. metadata-eval 75.0%

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

      11. metadata-eval 75.0%

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

      12. associate--l+ 75.0%

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

      13. fma-define 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in z around inf 87.8%

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

      1. associate--l+ 87.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 87.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 98.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 98.8%

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

      5. distribute-neg-frac 98.8%

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

      6. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \frac{x}{{z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 69.0%

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

      2. pow2 69.0%

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

      3. times-frac 69.0%

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

   9. Applied egg-rr 98.8%

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

   if 9.9999999999999994e303 < (*.f64 z z)

   1. Initial program 57.2%

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

      1. remove-double-neg 57.2%

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

      2. distribute-lft-neg-out 57.2%

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

      3. distribute-frac-neg2 57.2%

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

      4. distribute-frac-neg 57.2%

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

      5. neg-mul-1 57.2%

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

      6. distribute-lft-neg-out 57.2%

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

      7. *-commutative 57.2%

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

      8. distribute-lft-neg-in 57.2%

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

      9. times-frac 57.2%

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

      10. metadata-eval 57.2%

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

      11. metadata-eval 57.2%

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

      12. associate--l+ 57.2%

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

      13. fma-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in z around inf 65.2%

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

      1. associate--l+ 65.2%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 65.2%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 66.7%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 66.7%

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

      5. distribute-neg-frac 66.7%

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

      6. metadata-eval 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in y around 0 65.2%

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

      1. *-commutative 65.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot {z}^{2}}}{y} \]

      2. associate-/l* 65.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right)} \]

      3. unpow2 65.2%

         \[\leadsto 0.5 \cdot \left(\left(\frac{\color{blue}{x \cdot x}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      4. associate-/l* 69.8%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{x \cdot \frac{x}{{z}^{2}}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      5. fmm-def 69.8%

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

      6. metadata-eval 69.8%

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

   10. Simplified 69.8%

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

       1. div-inv 69.8%

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

       2. pow2 69.8%

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

       3. associate-*l* 74.2%

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

   12. Applied egg-rr 74.2%

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

       1. *-un-lft-identity 74.2%

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

       2. pow2 74.2%

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

       3. times-frac 89.5%

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

   14. Applied egg-rr 89.5%

       \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(x, \color{blue}{\frac{1}{z} \cdot \frac{x}{z}}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{z} \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq 10^{-64}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left({z}^{2} \cdot \left(\mathsf{fma}\left(\frac{x}{y}, t\_0, \frac{-1}{y}\right) + \frac{1}{z} \cdot \frac{y}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, t\_0, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 z) (/ x z))))
   (if (<= z 1e-64)
     (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
     (if (<= z 2.2e+161)
       (*
        0.5
        (* (pow z 2.0) (+ (fma (/ x y) t_0 (/ -1.0 y)) (* (/ 1.0 z) (/ y z)))))
       (* 0.5 (* (fma x t_0 -1.0) (* z (* z (/ 1.0 y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 / z) * (x / z);
	double tmp;
	if (z <= 1e-64) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (z <= 2.2e+161) {
		tmp = 0.5 * (pow(z, 2.0) * (fma((x / y), t_0, (-1.0 / y)) + ((1.0 / z) * (y / z))));
	} else {
		tmp = 0.5 * (fma(x, t_0, -1.0) * (z * (z * (1.0 / y))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 / z) * Float64(x / z))
	tmp = 0.0
	if (z <= 1e-64)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (z <= 2.2e+161)
		tmp = Float64(0.5 * Float64((z ^ 2.0) * Float64(fma(Float64(x / y), t_0, Float64(-1.0 / y)) + Float64(Float64(1.0 / z) * Float64(y / z)))));
	else
		tmp = Float64(0.5 * Float64(fma(x, t_0, -1.0) * Float64(z * Float64(z * Float64(1.0 / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1e-64], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.2e+161], N[(0.5 * N[(N[Power[z, 2.0], $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] * t$95$0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x * t$95$0 + -1.0), $MachinePrecision] * N[(z * N[(z * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 9.99999999999999965e-65

   1. Initial program 71.9%

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

   if 9.99999999999999965e-65 < z < 2.2e161

   1. Initial program 76.3%

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

      1. remove-double-neg 76.3%

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

      2. distribute-lft-neg-out 76.3%

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

      3. distribute-frac-neg2 76.3%

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

      4. distribute-frac-neg 76.3%

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

      5. neg-mul-1 76.3%

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

      6. distribute-lft-neg-out 76.3%

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

      7. *-commutative 76.3%

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

      8. distribute-lft-neg-in 76.3%

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

      9. times-frac 76.3%

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

      10. metadata-eval 76.3%

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

      11. metadata-eval 76.3%

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

      12. associate--l+ 76.3%

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

      13. fma-define 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in z around inf 87.4%

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

      1. associate--l+ 87.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 87.4%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 97.8%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 97.8%

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

      5. distribute-neg-frac 97.8%

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

      6. metadata-eval 97.8%

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

   7. Simplified 97.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \frac{x}{{z}^{2}}, \frac{-1}{y}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 74.1%

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

   9. Applied egg-rr 97.8%

      \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{1}{z} \cdot \frac{x}{z}}, \frac{-1}{y}\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-un-lft-identity 97.8%

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

       2. pow2 97.8%

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

       3. times-frac 97.7%

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

   11. Applied egg-rr 97.7%

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

   if 2.2e161 < z

   1. Initial program 67.1%

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

      1. remove-double-neg 67.1%

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

      2. distribute-lft-neg-out 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-frac-neg 67.1%

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

      5. neg-mul-1 67.1%

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

      6. distribute-lft-neg-out 67.1%

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

      7. *-commutative 67.1%

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

      8. distribute-lft-neg-in 67.1%

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

      9. times-frac 67.1%

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

      10. metadata-eval 67.1%

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

      11. metadata-eval 67.1%

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

      12. associate--l+ 67.1%

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

      13. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in z around inf 72.7%

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

      1. associate--l+ 72.7%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 72.7%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 69.9%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 69.9%

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

      5. distribute-neg-frac 69.9%

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

      6. metadata-eval 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in y around 0 72.7%

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

      1. *-commutative 72.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot {z}^{2}}}{y} \]

      2. associate-/l* 72.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right)} \]

      3. unpow2 72.7%

         \[\leadsto 0.5 \cdot \left(\left(\frac{\color{blue}{x \cdot x}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      4. associate-/l* 75.5%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{x \cdot \frac{x}{{z}^{2}}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      5. fmm-def 75.5%

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

      6. metadata-eval 75.5%

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

   10. Simplified 75.5%

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

       1. div-inv 75.5%

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

       2. pow2 75.5%

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

       3. associate-*l* 80.5%

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

   12. Applied egg-rr 80.5%

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

       1. *-un-lft-identity 80.5%

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

       2. pow2 80.5%

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

       3. times-frac 94.4%

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

   14. Applied egg-rr 94.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-64}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;0.5 \cdot \left({z}^{2} \cdot \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{z} \cdot \frac{x}{z}, \frac{-1}{y}\right) + \frac{1}{z} \cdot \frac{y}{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, \frac{1}{z} \cdot \frac{x}{z}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, \frac{1}{z} \cdot \frac{x}{z}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.6e-241)
   (* 0.5 (* (fma x (* (/ 1.0 z) (/ x z)) -1.0) (* z (* z (/ 1.0 y)))))
   (if (<= y 1.35e+154)
     (* 0.5 (/ (fma x x (- (* y y) (* z z))) y))
     (* y 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.6e-241) {
		tmp = 0.5 * (fma(x, ((1.0 / z) * (x / z)), -1.0) * (z * (z * (1.0 / y))));
	} else if (y <= 1.35e+154) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z * z))) / y);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.6e-241)
		tmp = Float64(0.5 * Float64(fma(x, Float64(Float64(1.0 / z) * Float64(x / z)), -1.0) * Float64(z * Float64(z * Float64(1.0 / y)))));
	elseif (y <= 1.35e+154)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.6e-241], N[(0.5 * N[(N[(x * N[(N[(1.0 / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(z * N[(z * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5999999999999997e-241

   1. Initial program 74.8%

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

      1. remove-double-neg 74.8%

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

      2. distribute-lft-neg-out 74.8%

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

      3. distribute-frac-neg2 74.8%

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

      4. distribute-frac-neg 74.8%

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

      5. neg-mul-1 74.8%

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

      6. distribute-lft-neg-out 74.8%

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

      7. *-commutative 74.8%

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

      8. distribute-lft-neg-in 74.8%

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

      9. times-frac 74.8%

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

      10. metadata-eval 74.8%

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

      11. metadata-eval 74.8%

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

      12. associate--l+ 74.8%

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

      13. fma-define 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in z around inf 55.5%

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

      1. associate--l+ 55.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 55.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 61.1%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 61.1%

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

      5. distribute-neg-frac 61.1%

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

      6. metadata-eval 61.1%

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

   7. Simplified 61.1%

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

   8. Taylor expanded in y around 0 50.1%

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

      1. *-commutative 50.1%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot {z}^{2}}}{y} \]

      2. associate-/l* 50.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right)} \]

      3. unpow2 50.1%

         \[\leadsto 0.5 \cdot \left(\left(\frac{\color{blue}{x \cdot x}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      4. associate-/l* 51.7%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{x \cdot \frac{x}{{z}^{2}}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      5. fmm-def 51.7%

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

      6. metadata-eval 51.7%

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

   10. Simplified 51.7%

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

       1. div-inv 51.7%

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

       2. pow2 51.7%

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

       3. associate-*l* 55.7%

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

   12. Applied egg-rr 55.7%

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

       1. *-un-lft-identity 55.7%

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

       2. pow2 55.7%

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

       3. times-frac 64.5%

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

   14. Applied egg-rr 64.5%

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

   if 8.5999999999999997e-241 < y < 1.35000000000000003e154

   1. Initial program 91.7%

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

      1. remove-double-neg 91.7%

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

      2. distribute-lft-neg-out 91.7%

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

      3. distribute-frac-neg2 91.7%

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

      4. distribute-frac-neg 91.7%

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

      5. neg-mul-1 91.7%

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

      6. distribute-lft-neg-out 91.7%

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

      7. *-commutative 91.7%

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

      8. distribute-lft-neg-in 91.7%

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

      9. times-frac 91.7%

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

      10. metadata-eval 91.7%

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

      11. metadata-eval 91.7%

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

      12. associate--l+ 91.7%

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

      13. fma-define 93.8%

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

   3. Simplified 93.8%

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

   if 1.35000000000000003e154 < y

   1. Initial program 14.8%

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

      1. remove-double-neg 14.8%

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

      2. distribute-lft-neg-out 14.8%

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

      3. distribute-frac-neg2 14.8%

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

      4. distribute-frac-neg 14.8%

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

      5. neg-mul-1 14.8%

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

      6. distribute-lft-neg-out 14.8%

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

      7. *-commutative 14.8%

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

      8. distribute-lft-neg-in 14.8%

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

      9. times-frac 14.8%

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

      10. metadata-eval 14.8%

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

      11. metadata-eval 14.8%

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

      12. associate--l+ 14.8%

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

      13. fma-define 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in y around inf 64.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-241}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(x, \frac{1}{z} \cdot \frac{x}{z}, -1\right) \cdot \left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+154) (* 0.5 (/ (fma x x (- (* y y) (* z z))) y)) (* y 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z * z))) / y);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+154], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 82.0%

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

      1. remove-double-neg 82.0%

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

      2. distribute-lft-neg-out 82.0%

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

      3. distribute-frac-neg2 82.0%

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

      4. distribute-frac-neg 82.0%

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

      5. neg-mul-1 82.0%

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

      6. distribute-lft-neg-out 82.0%

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

      7. *-commutative 82.0%

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

      8. distribute-lft-neg-in 82.0%

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

      9. times-frac 82.0%

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

      10. metadata-eval 82.0%

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

      11. metadata-eval 82.0%

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

      12. associate--l+ 82.0%

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

      13. fma-define 83.4%

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

   3. Simplified 83.4%

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

   if 1.35000000000000003e154 < y

   1. Initial program 14.8%

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

      1. remove-double-neg 14.8%

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

      2. distribute-lft-neg-out 14.8%

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

      3. distribute-frac-neg2 14.8%

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

      4. distribute-frac-neg 14.8%

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

      5. neg-mul-1 14.8%

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

      6. distribute-lft-neg-out 14.8%

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

      7. *-commutative 14.8%

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

      8. distribute-lft-neg-in 14.8%

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

      9. times-frac 14.8%

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

      10. metadata-eval 14.8%

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

      11. metadata-eval 14.8%

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

      12. associate--l+ 14.8%

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

      13. fma-define 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in y around inf 64.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-277}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-167}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* z (* z (/ -1.0 y))))))
   (if (<= x 3.5e-277)
     t_0
     (if (<= x 5.2e-167)
       (* y 0.5)
       (if (<= x 3.3e+72) t_0 (* x (* x (/ 0.5 y))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (z * (z * (-1.0 / y)));
	double tmp;
	if (x <= 3.5e-277) {
		tmp = t_0;
	} else if (x <= 5.2e-167) {
		tmp = y * 0.5;
	} else if (x <= 3.3e+72) {
		tmp = t_0;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (z * (z * ((-1.0d0) / y)))
    if (x <= 3.5d-277) then
        tmp = t_0
    else if (x <= 5.2d-167) then
        tmp = y * 0.5d0
    else if (x <= 3.3d+72) then
        tmp = t_0
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (z * (z * (-1.0 / y)));
	double tmp;
	if (x <= 3.5e-277) {
		tmp = t_0;
	} else if (x <= 5.2e-167) {
		tmp = y * 0.5;
	} else if (x <= 3.3e+72) {
		tmp = t_0;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (z * (z * (-1.0 / y)))
	tmp = 0
	if x <= 3.5e-277:
		tmp = t_0
	elif x <= 5.2e-167:
		tmp = y * 0.5
	elif x <= 3.3e+72:
		tmp = t_0
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(z * Float64(z * Float64(-1.0 / y))))
	tmp = 0.0
	if (x <= 3.5e-277)
		tmp = t_0;
	elseif (x <= 5.2e-167)
		tmp = Float64(y * 0.5);
	elseif (x <= 3.3e+72)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (z * (z * (-1.0 / y)));
	tmp = 0.0;
	if (x <= 3.5e-277)
		tmp = t_0;
	elseif (x <= 5.2e-167)
		tmp = y * 0.5;
	elseif (x <= 3.3e+72)
		tmp = t_0;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(z * N[(z * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5e-277], t$95$0, If[LessEqual[x, 5.2e-167], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 3.3e+72], t$95$0, N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.49999999999999983e-277 or 5.1999999999999998e-167 < x < 3.3e72

   1. Initial program 72.7%

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

      1. remove-double-neg 72.7%

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

      2. distribute-lft-neg-out 72.7%

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

      3. distribute-frac-neg2 72.7%

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

      4. distribute-frac-neg 72.7%

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

      5. neg-mul-1 72.7%

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

      6. distribute-lft-neg-out 72.7%

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

      7. *-commutative 72.7%

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

      8. distribute-lft-neg-in 72.7%

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

      9. times-frac 72.7%

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

      10. metadata-eval 72.7%

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

      11. metadata-eval 72.7%

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

      12. associate--l+ 72.7%

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

      13. fma-define 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in z around inf 57.6%

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

      1. associate--l+ 57.6%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \color{blue}{\left(\frac{y}{{z}^{2}} + \left(\frac{{x}^{2}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)}\right) \]

      2. unpow2 57.6%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\frac{\color{blue}{x \cdot x}}{y \cdot {z}^{2}} - \frac{1}{y}\right)\right)\right) \]

      3. times-frac 61.5%

         \[\leadsto 0.5 \cdot \left({z}^{2} \cdot \left(\frac{y}{{z}^{2}} + \left(\color{blue}{\frac{x}{y} \cdot \frac{x}{{z}^{2}}} - \frac{1}{y}\right)\right)\right) \]

      4. fmm-def 61.5%

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

      5. distribute-neg-frac 61.5%

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

      6. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in y around 0 49.5%

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

      1. *-commutative 49.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot {z}^{2}}}{y} \]

      2. associate-/l* 49.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\frac{{x}^{2}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right)} \]

      3. unpow2 49.4%

         \[\leadsto 0.5 \cdot \left(\left(\frac{\color{blue}{x \cdot x}}{{z}^{2}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      4. associate-/l* 50.4%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{x \cdot \frac{x}{{z}^{2}}} - 1\right) \cdot \frac{{z}^{2}}{y}\right) \]

      5. fmm-def 50.4%

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

      6. metadata-eval 50.4%

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

   10. Simplified 50.4%

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

       1. div-inv 50.4%

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

       2. pow2 50.4%

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

       3. associate-*l* 54.0%

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

   12. Applied egg-rr 54.0%

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

   13. Taylor expanded in x around 0 40.7%

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

   if 3.49999999999999983e-277 < x < 5.1999999999999998e-167

   1. Initial program 76.2%

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

      1. remove-double-neg 76.2%

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

      2. distribute-lft-neg-out 76.2%

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

      3. distribute-frac-neg2 76.2%

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

      4. distribute-frac-neg 76.2%

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

      5. neg-mul-1 76.2%

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

      6. distribute-lft-neg-out 76.2%

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

      7. *-commutative 76.2%

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

      8. distribute-lft-neg-in 76.2%

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

      9. times-frac 76.2%

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

      10. metadata-eval 76.2%

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

      11. metadata-eval 76.2%

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

      12. associate--l+ 76.2%

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

      13. fma-define 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around inf 63.3%

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

   if 3.3e72 < x

   1. Initial program 66.5%

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

      1. remove-double-neg 66.5%

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

      2. distribute-lft-neg-out 66.5%

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

      3. distribute-frac-neg2 66.5%

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

      4. distribute-frac-neg 66.5%

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

      5. neg-mul-1 66.5%

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

      6. distribute-lft-neg-out 66.5%

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

      7. *-commutative 66.5%

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

      8. distribute-lft-neg-in 66.5%

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

      9. times-frac 66.5%

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

      10. metadata-eval 66.5%

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

      11. metadata-eval 66.5%

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

      12. associate--l+ 66.5%

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

      13. fma-define 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.7%

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

      3. associate-*r/ 71.7%

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

   7. Simplified 71.7%

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

      1. pow2 71.7%

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

      2. associate-*r/ 71.7%

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

      3. clear-num 71.8%

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

      4. pow2 71.8%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}} \cdot 0.5}} \]

   9. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]
   10. Step-by-step derivation

       1. pow2 71.8%

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

       2. clear-num 71.7%

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

       3. associate-*r/ 71.7%

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

       4. *-commutative 71.7%

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

       5. associate-*r* 83.1%

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-167}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(z \cdot \frac{-1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+154) (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) (* y 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

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 (y <= 1.35d+154) then
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e+154:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e+154)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+154], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 82.0%

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

   if 1.35000000000000003e154 < y

   1. Initial program 14.8%

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

      1. remove-double-neg 14.8%

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

      2. distribute-lft-neg-out 14.8%

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

      3. distribute-frac-neg2 14.8%

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

      4. distribute-frac-neg 14.8%

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

      5. neg-mul-1 14.8%

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

      6. distribute-lft-neg-out 14.8%

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

      7. *-commutative 14.8%

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

      8. distribute-lft-neg-in 14.8%

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

      9. times-frac 14.8%

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

      10. metadata-eval 14.8%

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

      11. metadata-eval 14.8%

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

      12. associate--l+ 14.8%

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

      13. fma-define 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in y around inf 64.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+112) (* y 0.5) (* x (/ x (* y 2.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Fortran

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+112) then
        tmp = y * 0.5d0
    else
        tmp = x * (x / (y * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+112:
		tmp = y * 0.5
	else:
		tmp = x * (x / (y * 2.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+112)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+112)
		tmp = y * 0.5;
	else
		tmp = x * (x / (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+112], N[(y * 0.5), $MachinePrecision], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999993e111

   1. Initial program 74.6%

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

      1. remove-double-neg 74.6%

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

      2. distribute-lft-neg-out 74.6%

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

      3. distribute-frac-neg2 74.6%

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

      4. distribute-frac-neg 74.6%

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

      5. neg-mul-1 74.6%

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

      6. distribute-lft-neg-out 74.6%

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

      7. *-commutative 74.6%

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

      8. distribute-lft-neg-in 74.6%

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

      9. times-frac 74.6%

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

      10. metadata-eval 74.6%

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

      11. metadata-eval 74.6%

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

      12. associate--l+ 74.6%

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

      13. fma-define 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in y around inf 43.2%

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

   if 9.9999999999999993e111 < (*.f64 x x)

   1. Initial program 67.8%

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

      1. remove-double-neg 67.8%

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

      2. distribute-lft-neg-out 67.8%

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

      3. distribute-frac-neg2 67.8%

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

      4. distribute-frac-neg 67.8%

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

      5. neg-mul-1 67.8%

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

      6. distribute-lft-neg-out 67.8%

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

      7. *-commutative 67.8%

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

      8. distribute-lft-neg-in 67.8%

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

      9. times-frac 67.8%

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

      10. metadata-eval 67.8%

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

      11. metadata-eval 67.8%

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

      12. associate--l+ 67.8%

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

      13. fma-define 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in x around inf 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-*l/ 67.4%

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

      3. associate-*r/ 67.4%

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

   7. Simplified 67.4%

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

      1. pow2 67.4%

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

      2. associate-*r/ 67.4%

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

      3. clear-num 67.4%

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

      4. pow2 67.4%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}} \cdot 0.5}} \]

   9. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]
   10. Step-by-step derivation

       1. pow2 67.4%

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

       2. clear-num 67.4%

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

       3. associate-*r/ 67.4%

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

       4. associate-*l* 74.3%

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

       5. clear-num 74.3%

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

       6. div-inv 74.3%

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

       7. metadata-eval 74.3%

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

       8. add-sqr-sqrt 40.1%

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

       9. pow2 40.1%

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

       10. add-cbrt-cube 37.4%

           \[\leadsto x \cdot \left(x \cdot \frac{1}{{\color{blue}{\left(\sqrt[3]{\left(\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}\right) \cdot \sqrt{y \cdot 2}}\right)}}^{2}}\right) \]

       11. add-sqr-sqrt 37.4%

           \[\leadsto x \cdot \left(x \cdot \frac{1}{{\left(\sqrt[3]{\color{blue}{\left(y \cdot 2\right)} \cdot \sqrt{y \cdot 2}}\right)}^{2}}\right) \]

       12. cbrt-unprod 40.0%

           \[\leadsto x \cdot \left(x \cdot \frac{1}{{\color{blue}{\left(\sqrt[3]{y \cdot 2} \cdot \sqrt[3]{\sqrt{y \cdot 2}}\right)}}^{2}}\right) \]

       13. un-div-inv 40.0%

           \[\leadsto x \cdot \color{blue}{\frac{x}{{\left(\sqrt[3]{y \cdot 2} \cdot \sqrt[3]{\sqrt{y \cdot 2}}\right)}^{2}}} \]

       14. cbrt-unprod 37.4%

           \[\leadsto x \cdot \frac{x}{{\color{blue}{\left(\sqrt[3]{\left(y \cdot 2\right) \cdot \sqrt{y \cdot 2}}\right)}}^{2}} \]

       15. add-sqr-sqrt 37.3%

           \[\leadsto x \cdot \frac{x}{{\left(\sqrt[3]{\color{blue}{\left(\sqrt{y \cdot 2} \cdot \sqrt{y \cdot 2}\right)} \cdot \sqrt{y \cdot 2}}\right)}^{2}} \]

       16. add-cbrt-cube 40.1%

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

   11. Applied egg-rr 74.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+56}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e+56) (* y 0.5) (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+56) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

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 <= 2d+56) then
        tmp = y * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e+56) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2e+56:
		tmp = y * 0.5
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e+56)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2e+56)
		tmp = y * 0.5;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e+56], 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 < 2.00000000000000018e56

   1. Initial program 72.6%

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

      1. remove-double-neg 72.6%

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

      2. distribute-lft-neg-out 72.6%

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

      3. distribute-frac-neg2 72.6%

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

      4. distribute-frac-neg 72.6%

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

      5. neg-mul-1 72.6%

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

      6. distribute-lft-neg-out 72.6%

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

      7. *-commutative 72.6%

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

      8. distribute-lft-neg-in 72.6%

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

      9. times-frac 72.6%

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

      10. metadata-eval 72.6%

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

      11. metadata-eval 72.6%

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

      12. associate--l+ 72.6%

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

      13. fma-define 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in y around inf 36.5%

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

   if 2.00000000000000018e56 < x

   1. Initial program 69.0%

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

      1. remove-double-neg 69.0%

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

      2. distribute-lft-neg-out 69.0%

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

      3. distribute-frac-neg2 69.0%

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

      4. distribute-frac-neg 69.0%

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

      5. neg-mul-1 69.0%

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

      6. distribute-lft-neg-out 69.0%

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

      7. *-commutative 69.0%

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

      8. distribute-lft-neg-in 69.0%

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

      9. times-frac 69.0%

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

      10. metadata-eval 69.0%

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

      11. metadata-eval 69.0%

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

      12. associate--l+ 69.0%

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

      13. fma-define 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in x around inf 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*l/ 71.4%

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

      3. associate-*r/ 71.3%

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

   7. Simplified 71.3%

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

      1. pow2 71.3%

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

      2. associate-*r/ 71.4%

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

      3. clear-num 71.4%

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

      4. pow2 71.4%

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{{x}^{2}} \cdot 0.5}} \]

   9. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]
   10. Step-by-step derivation

       1. pow2 71.4%

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

       2. clear-num 71.4%

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

       3. associate-*r/ 71.3%

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

       4. *-commutative 71.3%

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

       5. associate-*r* 81.9%

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

   11. Applied egg-rr 81.9%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+56}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* y 0.5))

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

   1. remove-double-neg 72.0%

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

   2. distribute-lft-neg-out 72.0%

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

   3. distribute-frac-neg2 72.0%

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

   4. distribute-frac-neg 72.0%

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

   5. neg-mul-1 72.0%

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

   6. distribute-lft-neg-out 72.0%

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

   7. *-commutative 72.0%

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

   8. distribute-lft-neg-in 72.0%

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

   9. times-frac 72.0%

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

   10. metadata-eval 72.0%

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

   11. metadata-eval 72.0%

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

   12. associate--l+ 72.0%

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

   13. fma-define 73.2%

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

3. Simplified 73.2%

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

5. Taylor expanded in y around inf 32.5%

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

6. Final simplification 32.5%

   \[\leadsto y \cdot 0.5 \]
7. Add Preprocessing

Developer Target 1: 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 2024157 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

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