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

Percentage Accurate: 68.3% → 95.7%

Time: 10.1s

Alternatives: 7

Speedup: 1.1×

• 42.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% 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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y\_m + \mathsf{fma}\left(x, \frac{x}{y\_m}, z \cdot \frac{0}{y\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -5e-61)
      (* 0.5 (* (- x z) (/ z y_m)))
      (if (<= t_0 INFINITY)
        (* 0.5 (+ y_m (fma x (/ x y_m) (* z (/ 0.0 y_m)))))
        (* 0.5 (- y_m (* z (/ 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 * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -5e-61) {
		tmp = 0.5 * ((x - z) * (z / y_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (y_m + fma(x, (x / y_m), (z * (0.0 / y_m))));
	} else {
		tmp = 0.5 * (y_m - (z * (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(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-61)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(z / y_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * Float64(y_m + fma(x, Float64(x / y_m), Float64(z * Float64(0.0 / y_m)))));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(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[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-61], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision] + N[(z * N[(0.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.9999999999999999e-61

   1. Initial program 79.2%

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

   3. Taylor expanded in y around inf 86.1%

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

      1. *-commutative 86.1%

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

   5. Simplified 86.1%

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

      1. unpow2 86.1%

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

      2. unpow2 86.1%

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

      3. difference-of-squares 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in y around 0 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/l* 69.9%

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

      3. +-commutative 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in z around inf 40.6%

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

   if -4.9999999999999999e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 80.4%

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

      1. remove-double-neg 80.4%

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

      2. distribute-lft-neg-out 80.4%

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

      3. distribute-frac-neg2 80.4%

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

      4. distribute-frac-neg 80.4%

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

      5. neg-mul-1 80.4%

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

      6. distribute-lft-neg-out 80.4%

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

      7. *-commutative 80.4%

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

      8. distribute-lft-neg-in 80.4%

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

      9. times-frac 80.4%

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

      10. metadata-eval 80.4%

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

      11. metadata-eval 80.4%

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

      12. associate--l+ 80.4%

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

      13. fma-define 80.4%

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

   3. Simplified 80.4%

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

      1. prod-diff 66.7%

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

      2. fma-neg 66.7%

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

      3. difference-of-squares 66.7%

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

      4. fma-define 66.7%

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

      5. pow2 66.7%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in z around -inf 54.2%

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

   8. Taylor expanded in z around 0 64.2%

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

   10. Simplified 69.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(y + \mathsf{fma}\left(x, \frac{x}{y}, z \cdot \frac{0}{y}\right)\right)} \]

   if +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.0%

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

      1. remove-double-neg 0.0%

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

      2. distribute-lft-neg-out 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. distribute-frac-neg 0.0%

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

      5. neg-mul-1 0.0%

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

      6. distribute-lft-neg-out 0.0%

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

      7. *-commutative 0.0%

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

      8. distribute-lft-neg-in 0.0%

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

      9. times-frac 0.6%

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

      10. metadata-eval 0.6%

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

      11. metadata-eval 0.6%

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

      12. associate--l+ 0.6%

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

      13. fma-define 24.7%

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

   3. Simplified 24.7%

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

      1. prod-diff 0.6%

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

      2. fma-neg 0.6%

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

      3. difference-of-squares 2.4%

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

      4. fma-define 23.1%

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

      5. pow2 23.1%

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

   6. Applied egg-rr 23.1%

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

   7. Taylor expanded in z around -inf 22.5%

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

   8. Taylor expanded in z around 0 62.0%

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

   9. Taylor expanded in x around 0 79.3%

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

       1. neg-mul-1 79.3%

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

       2. +-commutative 79.3%

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

       3. distribute-rgt1-in 79.3%

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

       4. metadata-eval 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0} \cdot y}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       5. mul0-lft 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       6. associate-*r/ 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{\frac{-1 \cdot 0}{y}} + \left(-\frac{z}{y}\right)\right)\right) \]

       7. metadata-eval 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       8. unsub-neg 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \color{blue}{\left(\frac{0}{y} - \frac{z}{y}\right)}\right) \]

       9. div0 79.3%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{0} - \frac{z}{y}\right)\right) \]

       10. neg-sub0 79.3%

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

       11. *-commutative 79.3%

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

       12. distribute-lft-neg-out 79.3%

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

       13. distribute-rgt-neg-in 79.3%

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

   11. Simplified 79.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + \mathsf{fma}\left(x, \frac{x}{y}, z \cdot \frac{0}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10^{-150}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1e-150)
    (* 0.5 (* (- x z) (/ (+ x z) y_m)))
    (if (<= y_m 1.42e+141)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* 0.5 (- y_m (* z (/ 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 tmp;
	if (y_m <= 1e-150) {
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	} else if (y_m <= 1.42e+141) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (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) :: tmp
    if (y_m <= 1d-150) then
        tmp = 0.5d0 * ((x - z) * ((x + z) / y_m))
    else if (y_m <= 1.42d+141) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * (y_m - (z * (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 tmp;
	if (y_m <= 1e-150) {
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	} else if (y_m <= 1.42e+141) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * (y_m - (z * (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):
	tmp = 0
	if y_m <= 1e-150:
		tmp = 0.5 * ((x - z) * ((x + z) / y_m))
	elif y_m <= 1.42e+141:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * (y_m - (z * (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)
	tmp = 0.0
	if (y_m <= 1e-150)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(Float64(x + z) / y_m)));
	elseif (y_m <= 1.42e+141)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * 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)
	tmp = 0.0;
	if (y_m <= 1e-150)
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	elseif (y_m <= 1.42e+141)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * (y_m - (z * (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_] := N[(y$95$s * If[LessEqual[y$95$m, 1e-150], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.42e+141], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.00000000000000001e-150

   1. Initial program 75.5%

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

   3. Taylor expanded in y around inf 76.4%

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

      1. *-commutative 76.4%

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

   5. Simplified 76.4%

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

      1. unpow2 76.4%

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

      2. unpow2 76.4%

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

      3. difference-of-squares 82.9%

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in y around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-/l* 76.6%

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

      3. +-commutative 76.6%

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

   10. Simplified 76.6%

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

   if 1.00000000000000001e-150 < y < 1.42000000000000005e141

   1. Initial program 94.2%

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

   if 1.42000000000000005e141 < y

   1. Initial program 12.9%

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

      1. remove-double-neg 12.9%

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

      2. distribute-lft-neg-out 12.9%

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

      3. distribute-frac-neg2 12.9%

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

      4. distribute-frac-neg 12.9%

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

      5. neg-mul-1 12.9%

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

      6. distribute-lft-neg-out 12.9%

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

      7. *-commutative 12.9%

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

      8. distribute-lft-neg-in 12.9%

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

      9. times-frac 13.4%

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

      10. metadata-eval 13.4%

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

      11. metadata-eval 13.4%

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

      12. associate--l+ 13.4%

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

      13. fma-define 13.4%

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

   3. Simplified 13.4%

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

      1. prod-diff 13.4%

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

      2. fma-neg 13.4%

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

      3. difference-of-squares 14.1%

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

      4. fma-define 14.2%

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

      5. pow2 14.2%

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

   6. Applied egg-rr 14.2%

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

   7. Taylor expanded in z around -inf 9.8%

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

   8. Taylor expanded in z around 0 86.8%

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

   9. Taylor expanded in x around 0 87.7%

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

       1. neg-mul-1 87.7%

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

       2. +-commutative 87.7%

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

       3. distribute-rgt1-in 87.7%

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

       4. metadata-eval 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0} \cdot y}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       5. mul0-lft 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       6. associate-*r/ 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{\frac{-1 \cdot 0}{y}} + \left(-\frac{z}{y}\right)\right)\right) \]

       7. metadata-eval 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       8. unsub-neg 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \color{blue}{\left(\frac{0}{y} - \frac{z}{y}\right)}\right) \]

       9. div0 87.7%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{0} - \frac{z}{y}\right)\right) \]

       10. neg-sub0 87.7%

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

       11. *-commutative 87.7%

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

       12. distribute-lft-neg-out 87.7%

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

       13. distribute-rgt-neg-in 87.7%

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

   11. Simplified 87.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-150}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;y\_m \leq 1.55 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.5e-19)
    (* 0.5 (* (- x z) (/ z y_m)))
    (if (<= y_m 1.55e+102) (* 0.5 (* (- x z) (/ x y_m))) (* y_m 0.5)))))

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 <= 1.5e-19) {
		tmp = 0.5 * ((x - z) * (z / y_m));
	} else if (y_m <= 1.55e+102) {
		tmp = 0.5 * ((x - z) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 1.5d-19) then
        tmp = 0.5d0 * ((x - z) * (z / y_m))
    else if (y_m <= 1.55d+102) then
        tmp = 0.5d0 * ((x - z) * (x / y_m))
    else
        tmp = y_m * 0.5d0
    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 <= 1.5e-19) {
		tmp = 0.5 * ((x - z) * (z / y_m));
	} else if (y_m <= 1.55e+102) {
		tmp = 0.5 * ((x - z) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 1.5e-19:
		tmp = 0.5 * ((x - z) * (z / y_m))
	elif y_m <= 1.55e+102:
		tmp = 0.5 * ((x - z) * (x / y_m))
	else:
		tmp = y_m * 0.5
	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 <= 1.5e-19)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(z / y_m)));
	elseif (y_m <= 1.55e+102)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(x / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	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 <= 1.5e-19)
		tmp = 0.5 * ((x - z) * (z / y_m));
	elseif (y_m <= 1.55e+102)
		tmp = 0.5 * ((x - z) * (x / y_m));
	else
		tmp = y_m * 0.5;
	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, 1.5e-19], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.55e+102], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.49999999999999996e-19

   1. Initial program 78.8%

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

   3. Taylor expanded in y around inf 78.6%

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

      1. *-commutative 78.6%

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

   5. Simplified 78.6%

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

      1. unpow2 78.6%

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

      2. unpow2 78.6%

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

      3. difference-of-squares 84.2%

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

   7. Applied egg-rr 84.2%

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

   8. Taylor expanded in y around 0 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-/l* 77.9%

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

      3. +-commutative 77.9%

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

   10. Simplified 77.9%

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

   11. Taylor expanded in z around inf 49.8%

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

   if 1.49999999999999996e-19 < y < 1.54999999999999993e102

   1. Initial program 87.6%

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

   3. Taylor expanded in y around inf 87.7%

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

      1. *-commutative 87.7%

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

   5. Simplified 87.7%

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

      1. unpow2 87.7%

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

      2. unpow2 87.7%

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

      3. difference-of-squares 94.0%

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

   7. Applied egg-rr 94.0%

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

   8. Taylor expanded in y around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-/l* 87.5%

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

      3. +-commutative 87.5%

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

   10. Simplified 87.5%

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

   11. Taylor expanded in z around 0 63.6%

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

   if 1.54999999999999993e102 < y

   1. Initial program 30.0%

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

   3. Taylor expanded in y around inf 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

Alternative 4: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.35 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.35e+93)
    (* 0.5 (* (- x z) (/ (+ x z) y_m)))
    (* 0.5 (- y_m (* z (/ 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 tmp;
	if (y_m <= 3.35e+93) {
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	} else {
		tmp = 0.5 * (y_m - (z * (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) :: tmp
    if (y_m <= 3.35d+93) then
        tmp = 0.5d0 * ((x - z) * ((x + z) / y_m))
    else
        tmp = 0.5d0 * (y_m - (z * (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 tmp;
	if (y_m <= 3.35e+93) {
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	} else {
		tmp = 0.5 * (y_m - (z * (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):
	tmp = 0
	if y_m <= 3.35e+93:
		tmp = 0.5 * ((x - z) * ((x + z) / y_m))
	else:
		tmp = 0.5 * (y_m - (z * (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)
	tmp = 0.0
	if (y_m <= 3.35e+93)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(Float64(x + z) / y_m)));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * 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)
	tmp = 0.0;
	if (y_m <= 3.35e+93)
		tmp = 0.5 * ((x - z) * ((x + z) / y_m));
	else
		tmp = 0.5 * (y_m - (z * (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_] := N[(y$95$s * If[LessEqual[y$95$m, 3.35e+93], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.34999999999999983e93

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf 79.3%

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

      1. *-commutative 79.3%

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

   5. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

      3. difference-of-squares 85.0%

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in y around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/l* 78.7%

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

      3. +-commutative 78.7%

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

   10. Simplified 78.7%

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

   if 3.34999999999999983e93 < y

   1. Initial program 30.0%

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

      1. remove-double-neg 30.0%

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

      2. distribute-lft-neg-out 30.0%

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

      3. distribute-frac-neg2 30.0%

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

      4. distribute-frac-neg 30.0%

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

      5. neg-mul-1 30.0%

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

      6. distribute-lft-neg-out 30.0%

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

      7. *-commutative 30.0%

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

      8. distribute-lft-neg-in 30.0%

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

      9. times-frac 30.4%

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

      10. metadata-eval 30.4%

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

      11. metadata-eval 30.4%

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

      12. associate--l+ 30.4%

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

      13. fma-define 30.4%

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

   3. Simplified 30.4%

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

      1. prod-diff 30.4%

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

      2. fma-neg 30.4%

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

      3. difference-of-squares 31.0%

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

      4. fma-define 33.2%

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

      5. pow2 33.2%

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

   6. Applied egg-rr 33.2%

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

   7. Taylor expanded in z around -inf 14.6%

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

   8. Taylor expanded in z around 0 89.7%

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

   9. Taylor expanded in x around 0 81.9%

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

       1. neg-mul-1 81.9%

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

       2. +-commutative 81.9%

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

       3. distribute-rgt1-in 81.9%

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

       4. metadata-eval 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0} \cdot y}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       5. mul0-lft 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       6. associate-*r/ 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{\frac{-1 \cdot 0}{y}} + \left(-\frac{z}{y}\right)\right)\right) \]

       7. metadata-eval 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       8. unsub-neg 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \color{blue}{\left(\frac{0}{y} - \frac{z}{y}\right)}\right) \]

       9. div0 81.9%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{0} - \frac{z}{y}\right)\right) \]

       10. neg-sub0 81.9%

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

       11. *-commutative 81.9%

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

       12. distribute-lft-neg-out 81.9%

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

       13. distribute-rgt-neg-in 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.25 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 4.25e+89)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (* 0.5 (* (- x z) (/ x 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 (x <= 4.25e+89) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * ((x - z) * (x / 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 (x <= 4.25d+89) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = 0.5d0 * ((x - z) * (x / 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 (x <= 4.25e+89) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = 0.5 * ((x - z) * (x / 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 x <= 4.25e+89:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = 0.5 * ((x - z) * (x / 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 (x <= 4.25e+89)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(x / 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 (x <= 4.25e+89)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = 0.5 * ((x - z) * (x / 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[x, 4.25e+89], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.25000000000000023e89

   1. Initial program 71.2%

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

      1. remove-double-neg 71.2%

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

      2. distribute-lft-neg-out 71.2%

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

      3. distribute-frac-neg2 71.2%

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

      4. distribute-frac-neg 71.2%

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

      5. neg-mul-1 71.2%

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

      6. distribute-lft-neg-out 71.2%

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

      7. *-commutative 71.2%

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

      8. distribute-lft-neg-in 71.2%

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

      9. times-frac 71.3%

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

      10. metadata-eval 71.3%

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

      11. metadata-eval 71.3%

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

      12. associate--l+ 71.3%

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

      13. fma-define 73.7%

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

   3. Simplified 73.7%

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

      1. prod-diff 54.6%

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

      2. fma-neg 54.6%

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

      3. difference-of-squares 54.8%

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

      4. fma-define 56.2%

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

      5. pow2 56.2%

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

   6. Applied egg-rr 56.2%

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

   7. Taylor expanded in z around -inf 53.3%

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

   8. Taylor expanded in z around 0 90.7%

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

   9. Taylor expanded in x around 0 74.4%

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

       1. neg-mul-1 74.4%

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

       2. +-commutative 74.4%

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

       3. distribute-rgt1-in 74.4%

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

       4. metadata-eval 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0} \cdot y}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       5. mul0-lft 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(-1 \cdot \frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       6. associate-*r/ 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{\frac{-1 \cdot 0}{y}} + \left(-\frac{z}{y}\right)\right)\right) \]

       7. metadata-eval 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\frac{\color{blue}{0}}{y} + \left(-\frac{z}{y}\right)\right)\right) \]

       8. unsub-neg 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \color{blue}{\left(\frac{0}{y} - \frac{z}{y}\right)}\right) \]

       9. div0 74.4%

          \[\leadsto 0.5 \cdot \left(y + z \cdot \left(\color{blue}{0} - \frac{z}{y}\right)\right) \]

       10. neg-sub0 74.4%

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

       11. *-commutative 74.4%

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

       12. distribute-lft-neg-out 74.4%

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

       13. distribute-rgt-neg-in 74.4%

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

   11. Simplified 74.4%

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

   if 4.25000000000000023e89 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in y around inf 69.8%

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

      1. *-commutative 69.8%

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

   5. Simplified 69.8%

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

      1. unpow2 69.8%

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

      2. unpow2 69.8%

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

      3. difference-of-squares 80.6%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/l* 85.3%

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

      3. +-commutative 85.3%

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

   10. Simplified 85.3%

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

   11. Taylor expanded in z around 0 80.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.25 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.05 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 2.05e+94) (* 0.5 (* (- x z) (/ x y_m))) (* y_m 0.5))))

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 <= 2.05e+94) {
		tmp = 0.5 * ((x - z) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 2.05d+94) then
        tmp = 0.5d0 * ((x - z) * (x / y_m))
    else
        tmp = y_m * 0.5d0
    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 <= 2.05e+94) {
		tmp = 0.5 * ((x - z) * (x / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	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 <= 2.05e+94:
		tmp = 0.5 * ((x - z) * (x / y_m))
	else:
		tmp = y_m * 0.5
	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 <= 2.05e+94)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(x / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	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 <= 2.05e+94)
		tmp = 0.5 * ((x - z) * (x / y_m));
	else
		tmp = y_m * 0.5;
	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, 2.05e+94], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.05000000000000015e94

   1. Initial program 79.5%

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

   3. Taylor expanded in y around inf 79.3%

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

      1. *-commutative 79.3%

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

   5. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. unpow2 79.3%

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

      3. difference-of-squares 85.0%

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

   7. Applied egg-rr 85.0%

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

   8. Taylor expanded in y around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/l* 78.7%

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

      3. +-commutative 78.7%

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

   10. Simplified 78.7%

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

   11. Taylor expanded in z around 0 44.8%

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

   if 2.05000000000000015e94 < y

   1. Initial program 30.0%

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

   3. Taylor expanded in y around inf 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

Alternative 7: 34.2% accurate, 5.0× speedup

Math

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

FPCore

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

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 * (y_m * 0.5);
}

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 * (y_m * 0.5d0)
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 * (y_m * 0.5);
}

Python

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * 0.5))
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 * (y_m * 0.5);
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[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in y around inf 30.9%

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

   1. *-commutative 30.9%

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

5. Simplified 30.9%

   \[\leadsto \color{blue}{y \cdot 0.5} \]
6. 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 2024091 
(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)))