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

Percentage Accurate: 69.5% → 97.2%

Time: 13.0s

Alternatives: 11

Speedup: 0.9×

• 41.9% 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: 69.5% 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: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.15 \cdot 10^{-109}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y\_m} \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;y\_m \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{{y\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + x \cdot \frac{x}{y\_m}\right) - \frac{z}{\frac{y\_m}{z}}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.15e-109)
    (* (+ x z) (* (/ 0.5 y_m) (- x z)))
    (if (<= y_m 2.4e+23)
      (* y_m (+ 0.5 (* 0.5 (/ (* (+ x z) (- x z)) (pow y_m 2.0)))))
      (* 0.5 (- (+ y_m (* x (/ x y_m))) (/ z (/ y_m z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.15e-109) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else if (y_m <= 2.4e+23) {
		tmp = y_m * (0.5 + (0.5 * (((x + z) * (x - z)) / pow(y_m, 2.0))));
	} else {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.15d-109) then
        tmp = (x + z) * ((0.5d0 / y_m) * (x - z))
    else if (y_m <= 2.4d+23) then
        tmp = y_m * (0.5d0 + (0.5d0 * (((x + z) * (x - z)) / (y_m ** 2.0d0))))
    else
        tmp = 0.5d0 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.15e-109) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else if (y_m <= 2.4e+23) {
		tmp = y_m * (0.5 + (0.5 * (((x + z) * (x - z)) / Math.pow(y_m, 2.0))));
	} else {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.15e-109:
		tmp = (x + z) * ((0.5 / y_m) * (x - z))
	elif y_m <= 2.4e+23:
		tmp = y_m * (0.5 + (0.5 * (((x + z) * (x - z)) / math.pow(y_m, 2.0))))
	else:
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.15e-109)
		tmp = Float64(Float64(x + z) * Float64(Float64(0.5 / y_m) * Float64(x - z)));
	elseif (y_m <= 2.4e+23)
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x + z) * Float64(x - z)) / (y_m ^ 2.0)))));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x * Float64(x / y_m))) - Float64(z / Float64(y_m / z))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.15e-109)
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	elseif (y_m <= 2.4e+23)
		tmp = y_m * (0.5 + (0.5 * (((x + z) * (x - z)) / (y_m ^ 2.0))));
	else
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.15e-109], N[(N[(x + z), $MachinePrecision] * N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2.4e+23], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / N[Power[y$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 3.1500000000000001e-109

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0 65.2%

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

      1. associate-*r/ 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-/l* 65.3%

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

   5. Simplified 65.3%

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

      1. unpow2 65.3%

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

      2. unpow2 65.3%

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

      3. difference-of-squares 73.5%

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

   7. Applied egg-rr 73.5%

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

   8. Taylor expanded in y around 0 73.5%

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

      1. associate-*r/ 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-/l* 73.5%

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

      4. associate-*r* 77.7%

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

      5. +-commutative 77.7%

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

      6. metadata-eval 77.7%

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

      7. associate-*r/ 77.7%

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

      8. *-commutative 77.7%

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

      9. associate-*r/ 77.7%

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

      10. metadata-eval 77.7%

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

   10. Simplified 77.7%

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

   if 3.1500000000000001e-109 < y < 2.4e23

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf 85.9%

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

      1. *-commutative 85.9%

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

   5. Simplified 85.9%

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

      1. unpow2 65.9%

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

      2. unpow2 65.9%

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

      3. difference-of-squares 79.7%

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

   7. Applied egg-rr 99.7%

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

   if 2.4e23 < y

   1. Initial program 50.6%

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

      1. remove-double-neg 50.6%

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

      2. distribute-lft-neg-out 50.6%

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

      3. distribute-frac-neg2 50.6%

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

      4. distribute-frac-neg 50.6%

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

      5. neg-mul-1 50.6%

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

      6. distribute-lft-neg-out 50.6%

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

      7. *-commutative 50.6%

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

      8. distribute-lft-neg-in 50.6%

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

      9. times-frac 50.6%

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

      10. metadata-eval 50.6%

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

      11. metadata-eval 50.6%

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

      12. associate--l+ 50.6%

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

      13. fma-define 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in x around 0 78.9%

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

      1. div-inv 79.0%

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

      2. unpow2 79.0%

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

      3. associate-*l* 88.9%

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

   7. Applied egg-rr 88.9%

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

      1. unpow2 88.9%

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

      2. associate-/l* 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. add-sqr-sqrt 99.8%

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

       2. pow2 99.8%

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

       3. associate-*r* 88.5%

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

       4. unpow2 88.5%

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

       5. sqrt-prod 88.5%

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

       6. sqrt-pow1 99.8%

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

       7. metadata-eval 99.8%

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

       8. pow1 99.8%

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

       9. sqrt-div 99.7%

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

       10. metadata-eval 99.7%

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

       11. div-inv 99.8%

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

       12. pow2 99.8%

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

       13. clear-num 99.8%

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

       14. frac-times 99.8%

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

       15. *-un-lft-identity 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. rem-square-sqrt 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.15 \cdot 10^{-109}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y} \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{{y}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\frac{x}{\sqrt{y\_m}}, \sqrt{y\_m}\right)\\ t_1 := \frac{z}{\sqrt{y\_m}}\\ y\_s \cdot \left(0.5 \cdot \left(\left(t\_0 + t\_1\right) \cdot \left(t\_0 - t\_1\right)\right)\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (hypot (/ x (sqrt y_m)) (sqrt y_m))) (t_1 (/ z (sqrt y_m))))
   (* y_s (* 0.5 (* (+ t_0 t_1) (- t_0 t_1))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = hypot((x / sqrt(y_m)), sqrt(y_m));
	double t_1 = z / sqrt(y_m);
	return y_s * (0.5 * ((t_0 + t_1) * (t_0 - t_1)));
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = Math.hypot((x / Math.sqrt(y_m)), Math.sqrt(y_m));
	double t_1 = z / Math.sqrt(y_m);
	return y_s * (0.5 * ((t_0 + t_1) * (t_0 - t_1)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = math.hypot((x / math.sqrt(y_m)), math.sqrt(y_m))
	t_1 = z / math.sqrt(y_m)
	return y_s * (0.5 * ((t_0 + t_1) * (t_0 - t_1)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = hypot(Float64(x / sqrt(y_m)), sqrt(y_m))
	t_1 = Float64(z / sqrt(y_m))
	return Float64(y_s * Float64(0.5 * Float64(Float64(t_0 + t_1) * Float64(t_0 - t_1))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	t_0 = hypot((x / sqrt(y_m)), sqrt(y_m));
	t_1 = z / sqrt(y_m);
	tmp = y_s * (0.5 * ((t_0 + t_1) * (t_0 - t_1)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[Sqrt[N[(x / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] ^ 2 + N[Sqrt[y$95$m], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(z / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(0.5 * N[(N[(t$95$0 + t$95$1), $MachinePrecision] * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.6%

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

   1. remove-double-neg 66.6%

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

   2. distribute-lft-neg-out 66.6%

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

   3. distribute-frac-neg2 66.6%

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

   4. distribute-frac-neg 66.6%

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

   5. neg-mul-1 66.6%

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

   6. distribute-lft-neg-out 66.6%

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

   7. *-commutative 66.6%

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

   8. distribute-lft-neg-in 66.6%

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

   9. times-frac 66.6%

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

   10. metadata-eval 66.6%

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

   11. metadata-eval 66.6%

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

   12. associate--l+ 66.6%

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

   13. fma-define 70.5%

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

3. Simplified 70.5%

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

5. Taylor expanded in x around 0 77.9%

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

   1. add-sqr-sqrt 39.2%

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

   2. add-sqr-sqrt 39.2%

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

   3. difference-of-squares 39.2%

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

7. Applied egg-rr 49.3%

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

8. Final simplification 49.3%

   \[\leadsto 0.5 \cdot \left(\left(\mathsf{hypot}\left(\frac{x}{\sqrt{y}}, \sqrt{y}\right) + \frac{z}{\sqrt{y}}\right) \cdot \left(\mathsf{hypot}\left(\frac{x}{\sqrt{y}}, \sqrt{y}\right) - \frac{z}{\sqrt{y}}\right)\right) \]
9. Add Preprocessing

Alternative 3: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(x + z\right) \cdot \left(0.5 \cdot \frac{x}{y\_m}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.05 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (+ x z) (* 0.5 (/ x y_m)))))
   (*
    y_s
    (if (<= z 3.05e-265)
      (* 0.5 y_m)
      (if (<= z 3.3e-152)
        t_0
        (if (<= z 8.6e-57)
          (* 0.5 y_m)
          (if (<= z 5.6e+64)
            t_0
            (if (<= z 4e+90)
              (* 0.5 y_m)
              (if (<= z 1.7e+92)
                (* (/ 0.5 y_m) (* x x))
                (* (+ x z) (* -0.5 (/ z y_m))))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x + z) * (0.5 * (x / y_m));
	double tmp;
	if (z <= 3.05e-265) {
		tmp = 0.5 * y_m;
	} else if (z <= 3.3e-152) {
		tmp = t_0;
	} else if (z <= 8.6e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 5.6e+64) {
		tmp = t_0;
	} else if (z <= 4e+90) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.7e+92) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + z) * (0.5d0 * (x / y_m))
    if (z <= 3.05d-265) then
        tmp = 0.5d0 * y_m
    else if (z <= 3.3d-152) then
        tmp = t_0
    else if (z <= 8.6d-57) then
        tmp = 0.5d0 * y_m
    else if (z <= 5.6d+64) then
        tmp = t_0
    else if (z <= 4d+90) then
        tmp = 0.5d0 * y_m
    else if (z <= 1.7d+92) then
        tmp = (0.5d0 / y_m) * (x * x)
    else
        tmp = (x + z) * ((-0.5d0) * (z / y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x + z) * (0.5 * (x / y_m));
	double tmp;
	if (z <= 3.05e-265) {
		tmp = 0.5 * y_m;
	} else if (z <= 3.3e-152) {
		tmp = t_0;
	} else if (z <= 8.6e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 5.6e+64) {
		tmp = t_0;
	} else if (z <= 4e+90) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.7e+92) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (x + z) * (0.5 * (x / y_m))
	tmp = 0
	if z <= 3.05e-265:
		tmp = 0.5 * y_m
	elif z <= 3.3e-152:
		tmp = t_0
	elif z <= 8.6e-57:
		tmp = 0.5 * y_m
	elif z <= 5.6e+64:
		tmp = t_0
	elif z <= 4e+90:
		tmp = 0.5 * y_m
	elif z <= 1.7e+92:
		tmp = (0.5 / y_m) * (x * x)
	else:
		tmp = (x + z) * (-0.5 * (z / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x + z) * Float64(0.5 * Float64(x / y_m)))
	tmp = 0.0
	if (z <= 3.05e-265)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 3.3e-152)
		tmp = t_0;
	elseif (z <= 8.6e-57)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 5.6e+64)
		tmp = t_0;
	elseif (z <= 4e+90)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 1.7e+92)
		tmp = Float64(Float64(0.5 / y_m) * Float64(x * x));
	else
		tmp = Float64(Float64(x + z) * Float64(-0.5 * Float64(z / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (x + z) * (0.5 * (x / y_m));
	tmp = 0.0;
	if (z <= 3.05e-265)
		tmp = 0.5 * y_m;
	elseif (z <= 3.3e-152)
		tmp = t_0;
	elseif (z <= 8.6e-57)
		tmp = 0.5 * y_m;
	elseif (z <= 5.6e+64)
		tmp = t_0;
	elseif (z <= 4e+90)
		tmp = 0.5 * y_m;
	elseif (z <= 1.7e+92)
		tmp = (0.5 / y_m) * (x * x);
	else
		tmp = (x + z) * (-0.5 * (z / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + z), $MachinePrecision] * N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 3.05e-265], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 3.3e-152], t$95$0, If[LessEqual[z, 8.6e-57], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 5.6e+64], t$95$0, If[LessEqual[z, 4e+90], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 1.7e+92], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x + z), $MachinePrecision] * N[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 3.0500000000000001e-265 or 3.29999999999999998e-152 < z < 8.60000000000000043e-57 or 5.60000000000000047e64 < z < 3.99999999999999987e90

   1. Initial program 64.8%

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

   3. Taylor expanded in y around inf 40.4%

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

      1. *-commutative 40.4%

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

   5. Simplified 40.4%

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

   if 3.0500000000000001e-265 < z < 3.29999999999999998e-152 or 8.60000000000000043e-57 < z < 5.60000000000000047e64

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 70.1%

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

   5. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. difference-of-squares 70.1%

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

   7. Applied egg-rr 70.1%

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

   8. Taylor expanded in y around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 70.1%

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

      4. associate-*r* 70.1%

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

      5. +-commutative 70.1%

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

      6. metadata-eval 70.1%

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

      7. associate-*r/ 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r/ 70.1%

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

      10. metadata-eval 70.1%

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

   10. Simplified 70.1%

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

   11. Taylor expanded in x around inf 55.6%

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

   if 3.99999999999999987e90 < z < 1.6999999999999999e92

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   5. Simplified 67.0%

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

      1. add-cbrt-cube 66.8%

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

      2. pow3 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in x around inf 67.4%

      \[\leadsto \sqrt[3]{\color{blue}{{x}^{6}}} \cdot \frac{0.5}{y} \]
   9. Step-by-step derivation

      1. pow1/3 67.4%

         \[\leadsto \color{blue}{{\left({x}^{6}\right)}^{0.3333333333333333}} \cdot \frac{0.5}{y} \]

      2. pow-pow 67.4%

         \[\leadsto \color{blue}{{x}^{\left(6 \cdot 0.3333333333333333\right)}} \cdot \frac{0.5}{y} \]

      3. metadata-eval 67.4%

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

      4. unpow2 67.4%

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

   10. Applied egg-rr 67.4%

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

   if 1.6999999999999999e92 < z

   1. Initial program 51.7%

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

   3. Taylor expanded in y around 0 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-/l* 51.7%

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

   5. Simplified 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. difference-of-squares 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in y around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/l* 78.6%

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

      4. associate-*r* 84.9%

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

      5. +-commutative 84.9%

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

      6. metadata-eval 84.9%

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

      7. associate-*r/ 84.9%

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

      8. *-commutative 84.9%

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

      9. associate-*r/ 84.9%

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

      10. metadata-eval 84.9%

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

   10. Simplified 84.9%

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

   11. Taylor expanded in x around 0 74.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.05 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-152}:\\ \;\;\;\;\left(x + z\right) \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+64}:\\ \;\;\;\;\left(x + z\right) \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(x + z\right) \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (+ x z) (* x (/ 0.5 y_m)))))
   (*
    y_s
    (if (<= z 9e-266)
      (* 0.5 y_m)
      (if (<= z 1.4e-155)
        t_0
        (if (<= z 8.5e-57)
          (* 0.5 y_m)
          (if (<= z 8.5e+64)
            t_0
            (if (<= z 2.9e+89)
              (* 0.5 y_m)
              (if (<= z 1.7e+92)
                (* (/ 0.5 y_m) (* x x))
                (* (+ x z) (* -0.5 (/ z y_m))))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x + z) * (x * (0.5 / y_m));
	double tmp;
	if (z <= 9e-266) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.4e-155) {
		tmp = t_0;
	} else if (z <= 8.5e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 8.5e+64) {
		tmp = t_0;
	} else if (z <= 2.9e+89) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.7e+92) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + z) * (x * (0.5d0 / y_m))
    if (z <= 9d-266) then
        tmp = 0.5d0 * y_m
    else if (z <= 1.4d-155) then
        tmp = t_0
    else if (z <= 8.5d-57) then
        tmp = 0.5d0 * y_m
    else if (z <= 8.5d+64) then
        tmp = t_0
    else if (z <= 2.9d+89) then
        tmp = 0.5d0 * y_m
    else if (z <= 1.7d+92) then
        tmp = (0.5d0 / y_m) * (x * x)
    else
        tmp = (x + z) * ((-0.5d0) * (z / y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x + z) * (x * (0.5 / y_m));
	double tmp;
	if (z <= 9e-266) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.4e-155) {
		tmp = t_0;
	} else if (z <= 8.5e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 8.5e+64) {
		tmp = t_0;
	} else if (z <= 2.9e+89) {
		tmp = 0.5 * y_m;
	} else if (z <= 1.7e+92) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (x + z) * (x * (0.5 / y_m))
	tmp = 0
	if z <= 9e-266:
		tmp = 0.5 * y_m
	elif z <= 1.4e-155:
		tmp = t_0
	elif z <= 8.5e-57:
		tmp = 0.5 * y_m
	elif z <= 8.5e+64:
		tmp = t_0
	elif z <= 2.9e+89:
		tmp = 0.5 * y_m
	elif z <= 1.7e+92:
		tmp = (0.5 / y_m) * (x * x)
	else:
		tmp = (x + z) * (-0.5 * (z / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x + z) * Float64(x * Float64(0.5 / y_m)))
	tmp = 0.0
	if (z <= 9e-266)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 1.4e-155)
		tmp = t_0;
	elseif (z <= 8.5e-57)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 8.5e+64)
		tmp = t_0;
	elseif (z <= 2.9e+89)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 1.7e+92)
		tmp = Float64(Float64(0.5 / y_m) * Float64(x * x));
	else
		tmp = Float64(Float64(x + z) * Float64(-0.5 * Float64(z / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (x + z) * (x * (0.5 / y_m));
	tmp = 0.0;
	if (z <= 9e-266)
		tmp = 0.5 * y_m;
	elseif (z <= 1.4e-155)
		tmp = t_0;
	elseif (z <= 8.5e-57)
		tmp = 0.5 * y_m;
	elseif (z <= 8.5e+64)
		tmp = t_0;
	elseif (z <= 2.9e+89)
		tmp = 0.5 * y_m;
	elseif (z <= 1.7e+92)
		tmp = (0.5 / y_m) * (x * x);
	else
		tmp = (x + z) * (-0.5 * (z / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x + z), $MachinePrecision] * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 9e-266], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 1.4e-155], t$95$0, If[LessEqual[z, 8.5e-57], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 8.5e+64], t$95$0, If[LessEqual[z, 2.9e+89], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 1.7e+92], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x + z), $MachinePrecision] * N[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 9.0000000000000006e-266 or 1.4e-155 < z < 8.49999999999999955e-57 or 8.4999999999999998e64 < z < 2.90000000000000025e89

   1. Initial program 64.8%

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

   3. Taylor expanded in y around inf 40.4%

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

      1. *-commutative 40.4%

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

   5. Simplified 40.4%

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

   if 9.0000000000000006e-266 < z < 1.4e-155 or 8.49999999999999955e-57 < z < 8.4999999999999998e64

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 70.1%

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

   5. Simplified 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. difference-of-squares 70.1%

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

   7. Applied egg-rr 70.1%

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

   8. Taylor expanded in y around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 70.1%

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

      4. associate-*r* 70.1%

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

      5. +-commutative 70.1%

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

      6. metadata-eval 70.1%

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

      7. associate-*r/ 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r/ 70.1%

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

      10. metadata-eval 70.1%

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

   10. Simplified 70.1%

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

   11. Taylor expanded in x around inf 55.6%

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

       1. *-commutative 55.6%

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

       2. associate-*l/ 55.6%

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

       3. associate-*r/ 55.6%

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

   13. Simplified 55.6%

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

   if 2.90000000000000025e89 < z < 1.6999999999999999e92

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   5. Simplified 67.0%

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

      1. add-cbrt-cube 66.8%

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

      2. pow3 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in x around inf 67.4%

      \[\leadsto \sqrt[3]{\color{blue}{{x}^{6}}} \cdot \frac{0.5}{y} \]
   9. Step-by-step derivation

      1. pow1/3 67.4%

         \[\leadsto \color{blue}{{\left({x}^{6}\right)}^{0.3333333333333333}} \cdot \frac{0.5}{y} \]

      2. pow-pow 67.4%

         \[\leadsto \color{blue}{{x}^{\left(6 \cdot 0.3333333333333333\right)}} \cdot \frac{0.5}{y} \]

      3. metadata-eval 67.4%

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

      4. unpow2 67.4%

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

   10. Applied egg-rr 67.4%

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

   if 1.6999999999999999e92 < z

   1. Initial program 51.7%

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

   3. Taylor expanded in y around 0 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-/l* 51.7%

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

   5. Simplified 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. difference-of-squares 78.6%

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

   7. Applied egg-rr 78.6%

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

   8. Taylor expanded in y around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/l* 78.6%

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

      4. associate-*r* 84.9%

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

      5. +-commutative 84.9%

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

      6. metadata-eval 84.9%

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

      7. associate-*r/ 84.9%

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

      8. *-commutative 84.9%

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

      9. associate-*r/ 84.9%

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

      10. metadata-eval 84.9%

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

   10. Simplified 84.9%

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

   11. Taylor expanded in x around 0 74.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\left(x + z\right) \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+64}:\\ \;\;\;\;\left(x + z\right) \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{0.5}{y\_m} \cdot \left(x \cdot x\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.9 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 y_m) (* x x))))
   (*
    y_s
    (if (<= z 5.9e-265)
      (* 0.5 y_m)
      (if (<= z 2.1e-153)
        t_0
        (if (<= z 8e-57)
          (* 0.5 y_m)
          (if (<= z 9.2e+64)
            t_0
            (if (<= z 7.9e+93)
              (* 0.5 y_m)
              (* (+ x z) (* -0.5 (/ z y_m)))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (0.5 / y_m) * (x * x);
	double tmp;
	if (z <= 5.9e-265) {
		tmp = 0.5 * y_m;
	} else if (z <= 2.1e-153) {
		tmp = t_0;
	} else if (z <= 8e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 9.2e+64) {
		tmp = t_0;
	} else if (z <= 7.9e+93) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / y_m) * (x * x)
    if (z <= 5.9d-265) then
        tmp = 0.5d0 * y_m
    else if (z <= 2.1d-153) then
        tmp = t_0
    else if (z <= 8d-57) then
        tmp = 0.5d0 * y_m
    else if (z <= 9.2d+64) then
        tmp = t_0
    else if (z <= 7.9d+93) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x + z) * ((-0.5d0) * (z / y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (0.5 / y_m) * (x * x);
	double tmp;
	if (z <= 5.9e-265) {
		tmp = 0.5 * y_m;
	} else if (z <= 2.1e-153) {
		tmp = t_0;
	} else if (z <= 8e-57) {
		tmp = 0.5 * y_m;
	} else if (z <= 9.2e+64) {
		tmp = t_0;
	} else if (z <= 7.9e+93) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x + z) * (-0.5 * (z / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (0.5 / y_m) * (x * x)
	tmp = 0
	if z <= 5.9e-265:
		tmp = 0.5 * y_m
	elif z <= 2.1e-153:
		tmp = t_0
	elif z <= 8e-57:
		tmp = 0.5 * y_m
	elif z <= 9.2e+64:
		tmp = t_0
	elif z <= 7.9e+93:
		tmp = 0.5 * y_m
	else:
		tmp = (x + z) * (-0.5 * (z / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(0.5 / y_m) * Float64(x * x))
	tmp = 0.0
	if (z <= 5.9e-265)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 2.1e-153)
		tmp = t_0;
	elseif (z <= 8e-57)
		tmp = Float64(0.5 * y_m);
	elseif (z <= 9.2e+64)
		tmp = t_0;
	elseif (z <= 7.9e+93)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x + z) * Float64(-0.5 * Float64(z / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (0.5 / y_m) * (x * x);
	tmp = 0.0;
	if (z <= 5.9e-265)
		tmp = 0.5 * y_m;
	elseif (z <= 2.1e-153)
		tmp = t_0;
	elseif (z <= 8e-57)
		tmp = 0.5 * y_m;
	elseif (z <= 9.2e+64)
		tmp = t_0;
	elseif (z <= 7.9e+93)
		tmp = 0.5 * y_m;
	else
		tmp = (x + z) * (-0.5 * (z / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[z, 5.9e-265], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 2.1e-153], t$95$0, If[LessEqual[z, 8e-57], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[z, 9.2e+64], t$95$0, If[LessEqual[z, 7.9e+93], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x + z), $MachinePrecision] * N[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 5.90000000000000012e-265 or 2.10000000000000004e-153 < z < 7.99999999999999964e-57 or 9.2e64 < z < 7.8999999999999999e93

   1. Initial program 65.7%

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

   3. Taylor expanded in y around inf 40.1%

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

      1. *-commutative 40.1%

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

   5. Simplified 40.1%

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

   if 5.90000000000000012e-265 < z < 2.10000000000000004e-153 or 7.99999999999999964e-57 < z < 9.2e64

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-/l* 70.1%

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

   5. Simplified 70.1%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.8%

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

   7. Applied egg-rr 50.8%

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

   8. Taylor expanded in x around inf 40.7%

      \[\leadsto \sqrt[3]{\color{blue}{{x}^{6}}} \cdot \frac{0.5}{y} \]
   9. Step-by-step derivation

      1. pow1/3 40.2%

         \[\leadsto \color{blue}{{\left({x}^{6}\right)}^{0.3333333333333333}} \cdot \frac{0.5}{y} \]

      2. pow-pow 55.1%

         \[\leadsto \color{blue}{{x}^{\left(6 \cdot 0.3333333333333333\right)}} \cdot \frac{0.5}{y} \]

      3. metadata-eval 55.1%

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

      4. unpow2 55.1%

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

   10. Applied egg-rr 55.1%

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

   if 7.8999999999999999e93 < z

   1. Initial program 50.7%

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

   3. Taylor expanded in y around 0 50.7%

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

      1. associate-*r/ 50.7%

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

      2. *-commutative 50.7%

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

      3. associate-/l* 50.7%

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

   5. Simplified 50.7%

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

      1. unpow2 50.7%

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

      2. unpow2 50.7%

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

      3. difference-of-squares 78.1%

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

   7. Applied egg-rr 78.1%

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

   8. Taylor expanded in y around 0 78.1%

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

      1. associate-*r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-/l* 78.1%

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

      4. associate-*r* 84.6%

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

      5. +-commutative 84.6%

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

      6. metadata-eval 84.6%

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

      7. associate-*r/ 84.6%

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

      8. *-commutative 84.6%

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

      9. associate-*r/ 84.6%

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

      10. metadata-eval 84.6%

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

   10. Simplified 84.6%

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

   11. Taylor expanded in x around 0 74.3%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.9 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x + z\right) \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y\_m} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + x \cdot \frac{x}{y\_m}\right) - \frac{z}{\frac{y\_m}{z}}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.65e-118)
    (* (+ x z) (* (/ 0.5 y_m) (- x z)))
    (* 0.5 (- (+ y_m (* x (/ x y_m))) (/ z (/ y_m z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.65e-118) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.65d-118) then
        tmp = (x + z) * ((0.5d0 / y_m) * (x - z))
    else
        tmp = 0.5d0 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.65e-118) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else {
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.65e-118:
		tmp = (x + z) * ((0.5 / y_m) * (x - z))
	else:
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.65e-118)
		tmp = Float64(Float64(x + z) * Float64(Float64(0.5 / y_m) * Float64(x - z)));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x * Float64(x / y_m))) - Float64(z / Float64(y_m / z))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.65e-118)
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	else
		tmp = 0.5 * ((y_m + (x * (x / y_m))) - (z / (y_m / z)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.65e-118], N[(N[(x + z), $MachinePrecision] * N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e-118

   1. Initial program 69.6%

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

   3. Taylor expanded in y around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-/l* 65.1%

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

   5. Simplified 65.1%

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

      1. unpow2 65.1%

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

      2. unpow2 65.1%

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

      3. difference-of-squares 73.3%

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

   7. Applied egg-rr 73.3%

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

   8. Taylor expanded in y around 0 73.3%

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

      1. associate-*r/ 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-/l* 73.3%

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

      4. associate-*r* 77.5%

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

      5. +-commutative 77.5%

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

      6. metadata-eval 77.5%

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

      7. associate-*r/ 77.5%

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

      8. *-commutative 77.5%

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

      9. associate-*r/ 77.5%

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

      10. metadata-eval 77.5%

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

   10. Simplified 77.5%

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

   if 1.65e-118 < y

   1. Initial program 61.6%

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

      1. remove-double-neg 61.6%

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

      2. distribute-lft-neg-out 61.6%

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

      3. distribute-frac-neg2 61.6%

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

      4. distribute-frac-neg 61.6%

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

      5. neg-mul-1 61.6%

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

      6. distribute-lft-neg-out 61.6%

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

      7. *-commutative 61.6%

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

      8. distribute-lft-neg-in 61.6%

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

      9. times-frac 61.6%

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

      10. metadata-eval 61.6%

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

      11. metadata-eval 61.6%

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

      12. associate--l+ 61.6%

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

      13. fma-define 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in x around 0 81.2%

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

      1. div-inv 81.3%

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

      2. unpow2 81.3%

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

      3. associate-*l* 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. unpow2 88.1%

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

      2. associate-/l* 95.7%

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

   9. Applied egg-rr 95.7%

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

       1. add-sqr-sqrt 95.7%

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

       2. pow2 95.7%

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

       3. associate-*r* 87.9%

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

       4. unpow2 87.9%

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

       5. sqrt-prod 87.8%

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

       6. sqrt-pow1 95.7%

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

       7. metadata-eval 95.7%

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

       8. pow1 95.7%

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

       9. sqrt-div 95.6%

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

       10. metadata-eval 95.6%

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

       11. div-inv 95.7%

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

       12. pow2 95.7%

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

       13. clear-num 95.7%

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

       14. frac-times 95.7%

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

       15. *-un-lft-identity 95.7%

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

   11. Applied egg-rr 95.7%

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

       1. associate-*l/ 95.7%

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

       2. rem-square-sqrt 95.7%

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

   13. Simplified 95.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-118}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \frac{z}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(\left(x + z\right) \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5.4e+120) (* (/ 0.5 y_m) (* (+ x z) (- x z))) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.4e+120) {
		tmp = (0.5 / y_m) * ((x + z) * (x - z));
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 5.4d+120) then
        tmp = (0.5d0 / y_m) * ((x + z) * (x - z))
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5.4e+120) {
		tmp = (0.5 / y_m) * ((x + z) * (x - z));
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 5.4e+120:
		tmp = (0.5 / y_m) * ((x + z) * (x - z))
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5.4e+120)
		tmp = Float64(Float64(0.5 / y_m) * Float64(Float64(x + z) * Float64(x - z)));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5.4e+120)
		tmp = (0.5 / y_m) * ((x + z) * (x - z));
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5.4e+120], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.3999999999999999e120

   1. Initial program 73.5%

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

   3. Taylor expanded in y around 0 63.0%

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

      1. associate-*r/ 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. difference-of-squares 71.1%

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

   7. Applied egg-rr 71.1%

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

   if 5.3999999999999999e120 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf 79.5%

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

      1. *-commutative 79.5%

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

   5. Simplified 79.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(\left(x + z\right) \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y\_m} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 8.5e+122) (* (+ x z) (* (/ 0.5 y_m) (- x z))) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.5e+122) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 8.5d+122) then
        tmp = (x + z) * ((0.5d0 / y_m) * (x - z))
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.5e+122) {
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 8.5e+122:
		tmp = (x + z) * ((0.5 / y_m) * (x - z))
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 8.5e+122)
		tmp = Float64(Float64(x + z) * Float64(Float64(0.5 / y_m) * Float64(x - z)));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 8.5e+122)
		tmp = (x + z) * ((0.5 / y_m) * (x - z));
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 8.5e+122], N[(N[(x + z), $MachinePrecision] * N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000003e122

   1. Initial program 73.5%

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

   3. Taylor expanded in y around 0 63.0%

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

      1. associate-*r/ 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. difference-of-squares 71.1%

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

   7. Applied egg-rr 71.1%

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

   8. Taylor expanded in y around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-/l* 71.1%

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

      4. associate-*r* 75.9%

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

      5. +-commutative 75.9%

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

      6. metadata-eval 75.9%

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

      7. associate-*r/ 75.9%

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

      8. *-commutative 75.9%

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

      9. associate-*r/ 75.9%

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

      10. metadata-eval 75.9%

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

   10. Simplified 75.9%

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

   if 8.50000000000000003e122 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf 79.5%

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

      1. *-commutative 79.5%

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

   5. Simplified 79.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\left(x + z\right) \cdot \left(\frac{0.5}{y} \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{+121}:\\ \;\;\;\;\left(x + z\right) \cdot \frac{0.5 \cdot \left(x - z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.2e+121) (* (+ x z) (/ (* 0.5 (- x z)) y_m)) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e+121) {
		tmp = (x + z) * ((0.5 * (x - z)) / y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.2d+121) then
        tmp = (x + z) * ((0.5d0 * (x - z)) / y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e+121) {
		tmp = (x + z) * ((0.5 * (x - z)) / y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.2e+121:
		tmp = (x + z) * ((0.5 * (x - z)) / y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.2e+121)
		tmp = Float64(Float64(x + z) * Float64(Float64(0.5 * Float64(x - z)) / y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.2e+121)
		tmp = (x + z) * ((0.5 * (x - z)) / y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.2e+121], N[(N[(x + z), $MachinePrecision] * N[(N[(0.5 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e121

   1. Initial program 73.5%

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

   3. Taylor expanded in y around 0 63.0%

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

      1. associate-*r/ 63.0%

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

      2. *-commutative 63.0%

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

      3. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. difference-of-squares 71.1%

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

   7. Applied egg-rr 71.1%

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

   8. Taylor expanded in y around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-/l* 71.1%

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

      4. associate-*r* 75.9%

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

      5. +-commutative 75.9%

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

      6. metadata-eval 75.9%

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

      7. associate-*r/ 75.9%

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

      8. *-commutative 75.9%

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

      9. associate-*r/ 75.9%

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

      10. metadata-eval 75.9%

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

   10. Simplified 75.9%

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

       1. associate-*l/ 76.0%

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

   12. Applied egg-rr 76.0%

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

   if 1.2e121 < y

   1. Initial program 33.1%

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

   3. Taylor expanded in y around inf 79.5%

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

      1. *-commutative 79.5%

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

   5. Simplified 79.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+121}:\\ \;\;\;\;\left(x + z\right) \cdot \frac{0.5 \cdot \left(x - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 4.7e+27) (* (/ 0.5 y_m) (* x x)) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.7e+27) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4.7d+27) then
        tmp = (0.5d0 / y_m) * (x * x)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.7e+27) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 4.7e+27:
		tmp = (0.5 / y_m) * (x * x)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.7e+27)
		tmp = Float64(Float64(0.5 / y_m) * Float64(x * x));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 4.7e+27)
		tmp = (0.5 / y_m) * (x * x);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4.7e+27], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.69999999999999976e27

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.4%

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

   5. Simplified 65.4%

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

      1. add-cbrt-cube 53.5%

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

      2. pow3 53.5%

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

   7. Applied egg-rr 53.5%

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

   8. Taylor expanded in x around inf 29.0%

      \[\leadsto \sqrt[3]{\color{blue}{{x}^{6}}} \cdot \frac{0.5}{y} \]
   9. Step-by-step derivation

      1. pow1/3 28.9%

         \[\leadsto \color{blue}{{\left({x}^{6}\right)}^{0.3333333333333333}} \cdot \frac{0.5}{y} \]

      2. pow-pow 36.2%

         \[\leadsto \color{blue}{{x}^{\left(6 \cdot 0.3333333333333333\right)}} \cdot \frac{0.5}{y} \]

      3. metadata-eval 36.2%

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

      4. unpow2 36.2%

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

   10. Applied egg-rr 36.2%

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

   if 4.69999999999999976e27 < y

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf 66.3%

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

      1. *-commutative 66.3%

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

   5. Simplified 66.3%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* 0.5 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (0.5d0 * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (0.5 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (0.5 * y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in y around inf 34.3%

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

   1. *-commutative 34.3%

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

5. Simplified 34.3%

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

6. Final simplification 34.3%

   \[\leadsto 0.5 \cdot y \]
7. 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 2024080 
(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)))