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

Percentage Accurate: 69.1% → 91.6%

Time: 9.8s

Alternatives: 9

Speedup: 0.7×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y + z\right)\\ \mathbf{if}\;x \cdot x \leq 10^{+303}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \frac{t\_0}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot x (+ y z))))
   (if (<= (* x x) 1e+303)
     (* 0.5 (+ y (/ (- (pow x 2.0) (pow z 2.0)) y)))
     (* 0.5 (* t_0 (/ t_0 y))))))

C

double code(double x, double y, double z) {
	double t_0 = hypot(x, (y + z));
	double tmp;
	if ((x * x) <= 1e+303) {
		tmp = 0.5 * (y + ((pow(x, 2.0) - pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * (t_0 * (t_0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(x, (y + z));
	double tmp;
	if ((x * x) <= 1e+303) {
		tmp = 0.5 * (y + ((Math.pow(x, 2.0) - Math.pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * (t_0 * (t_0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.hypot(x, (y + z))
	tmp = 0
	if (x * x) <= 1e+303:
		tmp = 0.5 * (y + ((math.pow(x, 2.0) - math.pow(z, 2.0)) / y))
	else:
		tmp = 0.5 * (t_0 * (t_0 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = hypot(x, Float64(y + z))
	tmp = 0.0
	if (Float64(x * x) <= 1e+303)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x ^ 2.0) - (z ^ 2.0)) / y)));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(t_0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = hypot(x, (y + z));
	tmp = 0.0;
	if ((x * x) <= 1e+303)
		tmp = 0.5 * (y + (((x ^ 2.0) - (z ^ 2.0)) / y));
	else
		tmp = 0.5 * (t_0 * (t_0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y + z), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e+303], N[(0.5 * N[(y + N[(N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e303

   1. Initial program 78.7%

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

      1. remove-double-neg 78.7%

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

      2. distribute-lft-neg-out 78.7%

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

      3. distribute-frac-neg2 78.7%

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

      4. distribute-frac-neg 78.7%

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

      5. neg-mul-1 78.7%

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

      6. distribute-lft-neg-out 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-lft-neg-in 78.7%

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

      9. times-frac 78.7%

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

      10. metadata-eval 78.7%

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

      11. metadata-eval 78.7%

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

      12. associate--l+ 78.7%

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

      13. fma-define 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 92.5%

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

      1. associate--l+ 92.5%

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

      2. div-sub 97.8%

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

   7. Simplified 97.8%

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

   if 1e303 < (*.f64 x x)

   1. Initial program 55.1%

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

      1. remove-double-neg 55.1%

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

      2. distribute-lft-neg-out 55.1%

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

      3. distribute-frac-neg2 55.1%

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

      4. distribute-frac-neg 55.1%

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

      5. neg-mul-1 55.1%

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

      6. distribute-lft-neg-out 55.1%

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

      7. *-commutative 55.1%

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

      8. distribute-lft-neg-in 55.1%

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

      9. times-frac 55.3%

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

      10. metadata-eval 55.3%

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

      11. metadata-eval 55.3%

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

      12. associate--l+ 55.3%

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

      13. fma-define 65.5%

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

   3. Simplified 65.5%

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

      1. prod-diff 55.3%

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

      2. fma-neg 55.3%

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

      3. difference-of-squares 55.5%

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

      4. fma-define 68.5%

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

      5. pow2 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. add-sqr-sqrt 68.5%

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

      2. associate-/l* 68.5%

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

   8. Applied egg-rr 89.8%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+303}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{hypot}\left(x, y + z\right) \cdot \frac{\mathsf{hypot}\left(x, y + z\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.7 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.7e+152)
   (* 0.5 (+ y (/ (- (pow x 2.0) (pow z 2.0)) y)))
   (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.7e+152) {
		tmp = 0.5 * (y + ((pow(x, 2.0) - pow(z, 2.0)) / y));
	} 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 <= 6.7d+152) then
        tmp = 0.5d0 * (y + (((x ** 2.0d0) - (z ** 2.0d0)) / y))
    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 <= 6.7e+152) {
		tmp = 0.5 * (y + ((Math.pow(x, 2.0) - Math.pow(z, 2.0)) / y));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.7e+152:
		tmp = 0.5 * (y + ((math.pow(x, 2.0) - math.pow(z, 2.0)) / y))
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.7e+152)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x ^ 2.0) - (z ^ 2.0)) / y)));
	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 <= 6.7e+152)
		tmp = 0.5 * (y + (((x ^ 2.0) - (z ^ 2.0)) / y));
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.7e+152], N[(0.5 * N[(y + N[(N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.69999999999999987e152

   1. Initial program 77.3%

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

      1. remove-double-neg 77.3%

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

      2. distribute-lft-neg-out 77.3%

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

      3. distribute-frac-neg2 77.3%

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

      4. distribute-frac-neg 77.3%

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

      5. neg-mul-1 77.3%

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

      6. distribute-lft-neg-out 77.3%

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

      7. *-commutative 77.3%

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

      8. distribute-lft-neg-in 77.3%

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

      9. times-frac 77.3%

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

      10. metadata-eval 77.3%

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

      11. metadata-eval 77.3%

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

      12. associate--l+ 77.3%

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

      13. fma-define 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 87.4%

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

      1. associate--l+ 87.4%

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

      2. div-sub 94.0%

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

   7. Simplified 94.0%

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

   if 6.69999999999999987e152 < x

   1. Initial program 47.2%

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

      1. remove-double-neg 47.2%

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

      2. distribute-lft-neg-out 47.2%

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

      3. distribute-frac-neg2 47.2%

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

      4. distribute-frac-neg 47.2%

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

      5. neg-mul-1 47.2%

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

      6. distribute-lft-neg-out 47.2%

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

      7. *-commutative 47.2%

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

      8. distribute-lft-neg-in 47.2%

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

      9. times-frac 47.6%

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

      10. metadata-eval 47.6%

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

      11. metadata-eval 47.6%

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

      12. associate--l+ 47.6%

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

      13. fma-define 57.1%

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

   3. Simplified 57.1%

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

      1. clear-num 57.1%

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

      2. un-div-inv 57.1%

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

      3. fma-undefine 47.6%

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

      4. associate--l+ 47.6%

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

      5. add-sqr-sqrt 47.6%

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

      6. pow2 47.6%

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

      7. hypot-define 47.6%

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

      8. pow2 47.6%

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

   6. Applied egg-rr 47.6%

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

   7. Taylor expanded in x around inf 62.1%

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

      1. associate-/r/ 62.1%

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

      2. add-sqr-sqrt 44.8%

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

      3. sqrt-undiv 44.8%

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

      4. sqrt-undiv 44.8%

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

      5. pow2 44.8%

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

      6. unpow2 44.8%

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

      7. associate-*r* 53.3%

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

      8. pow2 53.3%

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

      9. sqrt-undiv 53.3%

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

      10. sqrt-undiv 53.3%

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

      11. add-sqr-sqrt 75.9%

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

   9. Applied egg-rr 75.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.7 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+120}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(2 \cdot z + \frac{{x}^{2}}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 5e+120) t_0 (* 0.5 (+ y (+ (* 2.0 z) (/ (pow x 2.0) y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 5e+120) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y + ((2.0 * z) + (pow(x, 2.0) / 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 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= 5d+120) then
        tmp = t_0
    else
        tmp = 0.5d0 * (y + ((2.0d0 * z) + ((x ** 2.0d0) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 5e+120) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y + ((2.0 * z) + (Math.pow(x, 2.0) / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 5e+120:
		tmp = t_0
	else:
		tmp = 0.5 * (y + ((2.0 * z) + (math.pow(x, 2.0) / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 5e+120)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(2.0 * z) + Float64((x ^ 2.0) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 5e+120)
		tmp = t_0;
	else
		tmp = 0.5 * (y + ((2.0 * z) + ((x ^ 2.0) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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[t$95$0, 5e+120], t$95$0, N[(0.5 * N[(y + N[(N[(2.0 * z), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.00000000000000019e120

   1. Initial program 88.2%

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

   if 5.00000000000000019e120 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 54.6%

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

      1. remove-double-neg 54.6%

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

      2. distribute-lft-neg-out 54.6%

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

      3. distribute-frac-neg2 54.6%

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

      4. distribute-frac-neg 54.6%

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

      5. neg-mul-1 54.6%

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

      6. distribute-lft-neg-out 54.6%

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

      7. *-commutative 54.6%

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

      8. distribute-lft-neg-in 54.6%

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

      9. times-frac 54.8%

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

      10. metadata-eval 54.8%

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

      11. metadata-eval 54.8%

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

      12. associate--l+ 54.8%

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

      13. fma-define 60.6%

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

   3. Simplified 60.6%

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

      1. prod-diff 44.9%

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

      2. fma-neg 44.9%

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

      3. difference-of-squares 45.1%

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

      4. fma-define 52.5%

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

      5. pow2 52.5%

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

   6. Applied egg-rr 52.5%

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

   8. Applied egg-rr 65.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\mathsf{hypot}\left(x, z + y\right)}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{\mathsf{hypot}\left(x, z + y\right)}{\sqrt[3]{y}}\right)} \]

   9. Taylor expanded in z around 0 58.8%

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

Alternative 4: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+145}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e+145) (* 0.5 (/ (fma x x (- (* y y) (* z z))) y)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+145) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z * z))) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e+145)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e+145], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1000000000000001e145

   1. Initial program 83.0%

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

      1. remove-double-neg 83.0%

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

      2. distribute-lft-neg-out 83.0%

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

      3. distribute-frac-neg2 83.0%

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

      4. distribute-frac-neg 83.0%

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

      5. neg-mul-1 83.0%

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

      6. distribute-lft-neg-out 83.0%

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

      7. *-commutative 83.0%

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

      8. distribute-lft-neg-in 83.0%

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

      9. times-frac 83.0%

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

      10. metadata-eval 83.0%

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

      11. metadata-eval 83.0%

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

      12. associate--l+ 83.0%

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

      13. fma-define 86.2%

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

   3. Simplified 86.2%

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

   if 4.1000000000000001e145 < y

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf 80.9%

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

Alternative 5: 43.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ t_1 := x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-292}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.085:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z (/ z y)) -0.5)) (t_1 (* x (* x (/ 0.5 y)))))
   (if (<= y 1.8e-292)
     t_0
     (if (<= y 5.4e-267)
       t_1
       (if (<= y 1.35e-169)
         t_0
         (if (<= y 3.2e-69) t_1 (if (<= y 0.085) t_0 (* 0.5 y))))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= 1.8e-292) {
		tmp = t_0;
	} else if (y <= 5.4e-267) {
		tmp = t_1;
	} else if (y <= 1.35e-169) {
		tmp = t_0;
	} else if (y <= 3.2e-69) {
		tmp = t_1;
	} else if (y <= 0.085) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = (z * (z / y)) * (-0.5d0)
    t_1 = x * (x * (0.5d0 / y))
    if (y <= 1.8d-292) then
        tmp = t_0
    else if (y <= 5.4d-267) then
        tmp = t_1
    else if (y <= 1.35d-169) then
        tmp = t_0
    else if (y <= 3.2d-69) then
        tmp = t_1
    else if (y <= 0.085d0) then
        tmp = t_0
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double t_1 = x * (x * (0.5 / y));
	double tmp;
	if (y <= 1.8e-292) {
		tmp = t_0;
	} else if (y <= 5.4e-267) {
		tmp = t_1;
	} else if (y <= 1.35e-169) {
		tmp = t_0;
	} else if (y <= 3.2e-69) {
		tmp = t_1;
	} else if (y <= 0.085) {
		tmp = t_0;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * (z / y)) * -0.5
	t_1 = x * (x * (0.5 / y))
	tmp = 0
	if y <= 1.8e-292:
		tmp = t_0
	elif y <= 5.4e-267:
		tmp = t_1
	elif y <= 1.35e-169:
		tmp = t_0
	elif y <= 3.2e-69:
		tmp = t_1
	elif y <= 0.085:
		tmp = t_0
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * Float64(z / y)) * -0.5)
	t_1 = Float64(x * Float64(x * Float64(0.5 / y)))
	tmp = 0.0
	if (y <= 1.8e-292)
		tmp = t_0;
	elseif (y <= 5.4e-267)
		tmp = t_1;
	elseif (y <= 1.35e-169)
		tmp = t_0;
	elseif (y <= 3.2e-69)
		tmp = t_1;
	elseif (y <= 0.085)
		tmp = t_0;
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * (z / y)) * -0.5;
	t_1 = x * (x * (0.5 / y));
	tmp = 0.0;
	if (y <= 1.8e-292)
		tmp = t_0;
	elseif (y <= 5.4e-267)
		tmp = t_1;
	elseif (y <= 1.35e-169)
		tmp = t_0;
	elseif (y <= 3.2e-69)
		tmp = t_1;
	elseif (y <= 0.085)
		tmp = t_0;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e-292], t$95$0, If[LessEqual[y, 5.4e-267], t$95$1, If[LessEqual[y, 1.35e-169], t$95$0, If[LessEqual[y, 3.2e-69], t$95$1, If[LessEqual[y, 0.085], t$95$0, N[(0.5 * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8000000000000001e-292 or 5.39999999999999975e-267 < y < 1.3500000000000001e-169 or 3.19999999999999999e-69 < y < 0.0850000000000000061

   1. Initial program 81.8%

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

   3. Taylor expanded in z around inf 41.2%

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

      1. *-commutative 41.2%

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

   5. Simplified 41.2%

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

      1. unpow2 41.2%

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

      2. associate-/l* 41.8%

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

   7. Applied egg-rr 41.8%

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

   if 1.8000000000000001e-292 < y < 5.39999999999999975e-267 or 1.3500000000000001e-169 < y < 3.19999999999999999e-69

   1. Initial program 93.9%

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

      1. remove-double-neg 93.9%

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

      2. distribute-lft-neg-out 93.9%

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

      3. distribute-frac-neg2 93.9%

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

      4. distribute-frac-neg 93.9%

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

      5. neg-mul-1 93.9%

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

      6. distribute-lft-neg-out 93.9%

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

      7. *-commutative 93.9%

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

      8. distribute-lft-neg-in 93.9%

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

      9. times-frac 93.9%

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

      10. metadata-eval 93.9%

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

      11. metadata-eval 93.9%

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

      12. associate--l+ 93.9%

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

      13. fma-define 96.9%

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

   3. Simplified 96.9%

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

      1. clear-num 96.7%

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

      2. un-div-inv 96.7%

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

      3. fma-undefine 93.7%

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

      4. associate--l+ 93.7%

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

      5. add-sqr-sqrt 93.7%

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

      6. pow2 93.7%

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

      7. hypot-define 93.7%

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

      8. pow2 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in x around inf 69.9%

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

      1. associate-/r/ 69.8%

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

      2. add-sqr-sqrt 69.7%

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

      3. sqrt-undiv 69.6%

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

      4. sqrt-undiv 69.5%

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

      5. pow2 69.5%

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

      6. unpow2 69.5%

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

      7. associate-*r* 69.5%

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

      8. pow2 69.5%

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

      9. sqrt-undiv 69.7%

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

      10. sqrt-undiv 69.6%

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

      11. add-sqr-sqrt 69.8%

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

   9. Applied egg-rr 69.8%

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

   if 0.0850000000000000061 < y

   1. Initial program 37.7%

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

   3. Taylor expanded in y around inf 69.1%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-292}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-267}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 0.085:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+145}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e+145) (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+145) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 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 (y <= 4.1d+145) then
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+145) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.1e+145:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e+145)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.1e+145)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e+145], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1000000000000001e145

   1. Initial program 83.0%

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

   if 4.1000000000000001e145 < y

   1. Initial program 12.9%

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

   3. Taylor expanded in y around inf 80.9%

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

Alternative 7: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.11:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.11) (* (* z (/ z y)) -0.5) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.11) {
		tmp = (z * (z / y)) * -0.5;
	} else {
		tmp = 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 (y <= 0.11d0) then
        tmp = (z * (z / y)) * (-0.5d0)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.11) {
		tmp = (z * (z / y)) * -0.5;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.11:
		tmp = (z * (z / y)) * -0.5
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.11)
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.11)
		tmp = (z * (z / y)) * -0.5;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.11], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.110000000000000001

   1. Initial program 83.9%

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

   3. Taylor expanded in z around inf 39.5%

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

      1. *-commutative 39.5%

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

   5. Simplified 39.5%

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

      1. unpow2 39.5%

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

      2. associate-/l* 39.9%

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

   7. Applied egg-rr 39.9%

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

   if 0.110000000000000001 < y

   1. Initial program 37.7%

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

   3. Taylor expanded in y around inf 69.1%

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

Alternative 8: 41.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 + (z / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

   1. remove-double-neg 72.3%

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

   2. distribute-lft-neg-out 72.3%

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

   3. distribute-frac-neg2 72.3%

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

   4. distribute-frac-neg 72.3%

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

   5. neg-mul-1 72.3%

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

   6. distribute-lft-neg-out 72.3%

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

   7. *-commutative 72.3%

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

   8. distribute-lft-neg-in 72.3%

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

   9. times-frac 72.4%

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

   10. metadata-eval 72.4%

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

   11. metadata-eval 72.4%

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

   12. associate--l+ 72.4%

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

   13. fma-define 75.1%

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

3. Simplified 75.1%

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

   1. prod-diff 61.9%

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

   2. fma-neg 61.9%

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

   3. difference-of-squares 62.0%

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

   4. fma-define 65.4%

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

   5. pow2 65.4%

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

6. Applied egg-rr 65.4%

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

8. Applied egg-rr 67.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\mathsf{hypot}\left(x, z + y\right)}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \frac{\mathsf{hypot}\left(x, z + y\right)}{\sqrt[3]{y}}\right)} \]

9. Taylor expanded in y around inf 41.5%

   \[\leadsto \color{blue}{y \cdot \left(0.5 + \frac{z}{y}\right)} \]
10. Add Preprocessing

Alternative 9: 35.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Taylor expanded in y around inf 34.9%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
4. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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