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

Percentage Accurate: 69.1% → 96.5%

Time: 10.6s

Alternatives: 10

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: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x + z\_m}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;z\_m \cdot \left(-0.5 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\frac{x - z\_m}{y\_m} \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (+ x z_m) y_m)))
   (*
    y_s
    (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -5e+155)
      (* z_m (* -0.5 t_0))
      (* y_m (+ 0.5 (* 0.5 (* (/ (- x z_m) y_m) t_0))))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (x + z_m) / y_m;
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -5e+155) {
		tmp = z_m * (-0.5 * t_0);
	} else {
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * t_0)));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + z_m) / y_m
    if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)) <= (-5d+155)) then
        tmp = z_m * ((-0.5d0) * t_0)
    else
        tmp = y_m * (0.5d0 + (0.5d0 * (((x - z_m) / y_m) * t_0)))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double t_0 = (x + z_m) / y_m;
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -5e+155) {
		tmp = z_m * (-0.5 * t_0);
	} else {
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * t_0)));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = (x + z_m) / y_m
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -5e+155:
		tmp = z_m * (-0.5 * t_0)
	else:
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * t_0)))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(Float64(x + z_m) / y_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -5e+155)
		tmp = Float64(z_m * Float64(-0.5 * t_0));
	else
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x - z_m) / y_m) * t_0))));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = (x + z_m) / y_m;
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -5e+155)
		tmp = z_m * (-0.5 * t_0);
	else
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * t_0)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := Block[{t$95$0 = N[(N[(x + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -5e+155], N[(z$95$m * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.9999999999999999e155

   1. Initial program 59.9%

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

      1. remove-double-neg 59.9%

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

      2. distribute-lft-neg-out 59.9%

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

      3. distribute-frac-neg2 59.9%

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

      4. distribute-frac-neg 59.9%

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

      5. neg-mul-1 59.9%

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

      6. distribute-lft-neg-out 59.9%

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

      7. *-commutative 59.9%

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

      8. distribute-lft-neg-in 59.9%

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

      9. times-frac 59.9%

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

      10. metadata-eval 59.9%

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

      11. metadata-eval 59.9%

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

      12. associate--l+ 59.9%

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

      13. fma-define 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in y around 0 58.6%

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

      1. associate-*r/ 58.6%

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

      2. *-commutative 58.6%

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

      3. associate-/l* 58.6%

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

   7. Simplified 58.6%

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

      1. pow2 58.6%

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

      2. pow2 58.6%

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

      3. difference-of-squares 58.6%

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

   9. Applied egg-rr 58.6%

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

   10. Taylor expanded in x around 0 23.7%

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

       1. neg-mul-1 23.7%

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

   12. Simplified 23.7%

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

   13. Taylor expanded in z around 0 28.7%

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

       1. distribute-lft-out 28.7%

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

       2. *-lft-identity 28.7%

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

       3. associate-*l/ 28.7%

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

       4. *-lft-identity 28.7%

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

       5. associate-*l/ 28.7%

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

       6. distribute-lft-in 29.8%

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

       7. *-commutative 29.8%

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

       8. associate-*r/ 29.8%

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

       9. *-rgt-identity 29.8%

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

       10. +-commutative 29.8%

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

   15. Simplified 29.8%

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

   if -4.9999999999999999e155 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 70.7%

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

      1. remove-double-neg 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. distribute-frac-neg2 70.7%

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

      4. distribute-frac-neg 70.7%

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

      5. neg-mul-1 70.7%

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

      6. distribute-lft-neg-out 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-lft-neg-in 70.7%

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

      9. times-frac 70.3%

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

      10. metadata-eval 70.3%

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

      11. metadata-eval 70.3%

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

      12. associate--l+ 70.3%

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

      13. fma-define 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in y around inf 72.1%

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

      1. pow2 51.1%

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

      2. pow2 51.1%

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

      3. difference-of-squares 57.9%

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

   7. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. unpow2 76.9%

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

      3. times-frac 94.3%

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

   9. Applied egg-rr 94.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x - z}{y} \cdot \frac{x + z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z\_m}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(x + z\_m\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z\_m}{y\_m} \cdot \frac{x}{y\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(t\_0 \cdot \frac{z\_m}{y\_m}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- x z_m) y_m))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      (* 0.5 (* t_0 (+ x z_m)))
      (if (<= t_1 1e+298)
        t_1
        (if (<= t_1 INFINITY)
          (* y_m (+ 0.5 (* 0.5 (* (/ (+ x z_m) y_m) (/ x y_m)))))
          (* y_m (+ 0.5 (* 0.5 (* t_0 (/ z_m y_m)))))))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (x - z_m) / y_m;
	double t_1 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 0.5 * (t_0 * (x + z_m));
	} else if (t_1 <= 1e+298) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = y_m * (0.5 + (0.5 * (t_0 * (z_m / y_m))));
	}
	return y_s * tmp;
}

Java

z_m = Math.abs(z);
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_m) {
	double t_0 = (x - z_m) / y_m;
	double t_1 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 0.5 * (t_0 * (x + z_m));
	} else if (t_1 <= 1e+298) {
		tmp = t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = y_m * (0.5 + (0.5 * (t_0 * (z_m / y_m))));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = (x - z_m) / y_m
	t_1 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = 0.5 * (t_0 * (x + z_m))
	elif t_1 <= 1e+298:
		tmp = t_1
	elif t_1 <= math.inf:
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))))
	else:
		tmp = y_m * (0.5 + (0.5 * (t_0 * (z_m / y_m))))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(Float64(x - z_m) / y_m)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(0.5 * Float64(t_0 * Float64(x + z_m)));
	elseif (t_1 <= 1e+298)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x + z_m) / y_m) * Float64(x / y_m)))));
	else
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(t_0 * Float64(z_m / y_m)))));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = (x - z_m) / y_m;
	t_1 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = 0.5 * (t_0 * (x + z_m));
	elseif (t_1 <= 1e+298)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	else
		tmp = y_m * (0.5 + (0.5 * (t_0 * (z_m / y_m))));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := Block[{t$95$0 = N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], N[(0.5 * N[(t$95$0 * N[(x + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+298], t$95$1, If[LessEqual[t$95$1, Infinity], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.5 + N[(0.5 * N[(t$95$0 * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 0.0

   1. Initial program 64.5%

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

      1. remove-double-neg 64.5%

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

      2. distribute-lft-neg-out 64.5%

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

      3. distribute-frac-neg2 64.5%

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

      4. distribute-frac-neg 64.5%

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

      5. neg-mul-1 64.5%

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

      6. distribute-lft-neg-out 64.5%

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

      7. *-commutative 64.5%

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

      8. distribute-lft-neg-in 64.5%

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

      9. times-frac 64.5%

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

      10. metadata-eval 64.5%

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

      11. metadata-eval 64.5%

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

      12. associate--l+ 64.5%

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

      13. fma-define 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in y around inf 77.8%

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

      1. pow2 51.5%

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

      2. pow2 51.5%

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

      3. difference-of-squares 51.5%

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

   7. Applied egg-rr 77.8%

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

   8. Taylor expanded in y around 0 51.5%

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

      1. associate-/l* 58.5%

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

   10. Simplified 58.5%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 9.9999999999999996e297

   1. Initial program 99.6%

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

   if 9.9999999999999996e297 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.0%

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

      1. remove-double-neg 69.0%

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

      2. distribute-lft-neg-out 69.0%

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

      3. distribute-frac-neg2 69.0%

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

      4. distribute-frac-neg 69.0%

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

      5. neg-mul-1 69.0%

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

      6. distribute-lft-neg-out 69.0%

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

      7. *-commutative 69.0%

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

      8. distribute-lft-neg-in 69.0%

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

      9. times-frac 67.8%

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

      10. metadata-eval 67.8%

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

      11. metadata-eval 67.8%

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

      12. associate--l+ 67.8%

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

      13. fma-define 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in y around inf 91.4%

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

      1. pow2 67.8%

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

      2. pow2 67.8%

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

      3. difference-of-squares 67.8%

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

   7. Applied egg-rr 91.4%

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

      1. *-commutative 91.4%

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

      2. unpow2 91.4%

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

      3. times-frac 98.8%

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

   9. Applied egg-rr 98.8%

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

   10. Taylor expanded in x around inf 76.0%

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

   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 0.0%

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

      10. metadata-eval 0.0%

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

      11. metadata-eval 0.0%

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

      12. associate--l+ 0.0%

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

      13. fma-define 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in y around inf 0.0%

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

      1. pow2 10.9%

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

      2. pow2 10.9%

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

      3. difference-of-squares 69.3%

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

   7. Applied egg-rr 42.1%

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

      1. *-commutative 42.1%

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

      2. unpow2 42.1%

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

      3. times-frac 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 79.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+298}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z}{y} \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x - z}{y} \cdot \frac{z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{+47}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z\_m}{y\_m} \cdot \frac{x}{y\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\frac{x - z\_m}{y\_m} \cdot \frac{z\_m}{y\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= z_m 7.5e+47)
    (* y_m (+ 0.5 (* 0.5 (* (/ (+ x z_m) y_m) (/ x y_m)))))
    (* y_m (+ 0.5 (* 0.5 (* (/ (- x z_m) y_m) (/ z_m y_m))))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 7.5e+47) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * (z_m / y_m))));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 7.5d+47) then
        tmp = y_m * (0.5d0 + (0.5d0 * (((x + z_m) / y_m) * (x / y_m))))
    else
        tmp = y_m * (0.5d0 + (0.5d0 * (((x - z_m) / y_m) * (z_m / y_m))))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (z_m <= 7.5e+47) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * (z_m / y_m))));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 7.5e+47:
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))))
	else:
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * (z_m / y_m))))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 7.5e+47)
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x + z_m) / y_m) * Float64(x / y_m)))));
	else
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x - z_m) / y_m) * Float64(z_m / y_m)))));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 7.5e+47)
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	else
		tmp = y_m * (0.5 + (0.5 * (((x - z_m) / y_m) * (z_m / y_m))));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 7.5e+47], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.4999999999999999e47

   1. Initial program 67.3%

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

      1. remove-double-neg 67.3%

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

      2. distribute-lft-neg-out 67.3%

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

      3. distribute-frac-neg2 67.3%

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

      4. distribute-frac-neg 67.3%

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

      5. neg-mul-1 67.3%

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

      6. distribute-lft-neg-out 67.3%

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

      7. *-commutative 67.3%

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

      8. distribute-lft-neg-in 67.3%

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

      9. times-frac 66.9%

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

      10. metadata-eval 66.9%

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

      11. metadata-eval 66.9%

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

      12. associate--l+ 66.9%

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

      13. fma-define 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in y around inf 78.4%

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

      1. pow2 51.3%

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

      2. pow2 51.3%

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

      3. difference-of-squares 53.7%

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

   7. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. times-frac 93.8%

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

   9. Applied egg-rr 93.8%

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

   10. Taylor expanded in x around inf 81.3%

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

   if 7.4999999999999999e47 < z

   1. Initial program 65.2%

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

      1. remove-double-neg 65.2%

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

      2. distribute-lft-neg-out 65.2%

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

      3. distribute-frac-neg2 65.2%

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

      4. distribute-frac-neg 65.2%

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

      5. neg-mul-1 65.2%

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

      6. distribute-lft-neg-out 65.2%

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

      7. *-commutative 65.2%

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

      8. distribute-lft-neg-in 65.2%

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

      9. times-frac 65.2%

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

      10. metadata-eval 65.2%

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

      11. metadata-eval 65.2%

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

      12. associate--l+ 65.2%

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

      13. fma-define 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y around inf 63.0%

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

      1. pow2 65.3%

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

      2. pow2 65.3%

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

      3. difference-of-squares 78.8%

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

   7. Applied egg-rr 71.9%

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

      1. *-commutative 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 95.7%

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

   9. Applied egg-rr 95.7%

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

   10. Taylor expanded in x around 0 87.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z}{y} \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x - z}{y} \cdot \frac{z}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.46 \cdot 10^{+34}:\\ \;\;\;\;y\_m \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z\_m}{y\_m} \cdot \frac{x}{y\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z\_m}{y\_m} \cdot \left(x + z\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= z_m 1.46e+34)
    (* y_m (+ 0.5 (* 0.5 (* (/ (+ x z_m) y_m) (/ x y_m)))))
    (* 0.5 (* (/ (- x z_m) y_m) (+ x z_m))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.46e+34) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.46d+34) then
        tmp = y_m * (0.5d0 + (0.5d0 * (((x + z_m) / y_m) * (x / y_m))))
    else
        tmp = 0.5d0 * (((x - z_m) / y_m) * (x + z_m))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (z_m <= 1.46e+34) {
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	} else {
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 1.46e+34:
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))))
	else:
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 1.46e+34)
		tmp = Float64(y_m * Float64(0.5 + Float64(0.5 * Float64(Float64(Float64(x + z_m) / y_m) * Float64(x / y_m)))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(x - z_m) / y_m) * Float64(x + z_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 1.46e+34)
		tmp = y_m * (0.5 + (0.5 * (((x + z_m) / y_m) * (x / y_m))));
	else
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 1.46e+34], N[(y$95$m * N[(0.5 + N[(0.5 * N[(N[(N[(x + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.46e34

   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 66.3%

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

      10. metadata-eval 66.3%

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

      11. metadata-eval 66.3%

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

      12. associate--l+ 66.3%

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

      13. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in y around inf 78.0%

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

      1. pow2 50.4%

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

      2. pow2 50.4%

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

      3. difference-of-squares 52.8%

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

   7. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

      2. unpow2 79.9%

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

      3. times-frac 93.7%

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

   9. Applied egg-rr 93.7%

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

   10. Taylor expanded in x around inf 81.1%

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

   if 1.46e34 < z

   1. Initial program 68.0%

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

      1. remove-double-neg 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-frac-neg2 68.0%

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

      4. distribute-frac-neg 68.0%

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

      5. neg-mul-1 68.0%

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

      6. distribute-lft-neg-out 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-lft-neg-in 68.0%

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

      9. times-frac 68.0%

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

      10. metadata-eval 68.0%

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

      11. metadata-eval 68.0%

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

      12. associate--l+ 68.0%

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

      13. fma-define 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around inf 66.0%

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

      1. pow2 68.1%

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

      2. pow2 68.1%

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

      3. difference-of-squares 80.6%

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

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in y around 0 80.5%

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

      1. associate-/l* 89.0%

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

   10. Simplified 89.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.46 \cdot 10^{+34}:\\ \;\;\;\;y \cdot \left(0.5 + 0.5 \cdot \left(\frac{x + z}{y} \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z\_m}{y\_m} \cdot \left(x + z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= y_m 4.2e+100) (* 0.5 (* (/ (- x z_m) y_m) (+ x z_m))) (* y_m 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (y_m <= 4.2e+100) {
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 4.2d+100) then
        tmp = 0.5d0 * (((x - z_m) / y_m) * (x + z_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (y_m <= 4.2e+100) {
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if y_m <= 4.2e+100:
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (y_m <= 4.2e+100)
		tmp = Float64(0.5 * Float64(Float64(Float64(x - z_m) / y_m) * Float64(x + z_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 4.2e+100)
		tmp = 0.5 * (((x - z_m) / y_m) * (x + z_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 4.2e+100], N[(0.5 * N[(N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999997e100

   1. Initial program 72.7%

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

      1. remove-double-neg 72.7%

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

      2. distribute-lft-neg-out 72.7%

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

      3. distribute-frac-neg2 72.7%

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

      4. distribute-frac-neg 72.7%

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

      5. neg-mul-1 72.7%

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

      6. distribute-lft-neg-out 72.7%

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

      7. *-commutative 72.7%

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

      8. distribute-lft-neg-in 72.7%

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

      9. times-frac 72.3%

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

      10. metadata-eval 72.3%

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

      11. metadata-eval 72.3%

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

      12. associate--l+ 72.3%

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

      13. fma-define 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y around inf 75.7%

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

      1. pow2 60.0%

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

      2. pow2 60.0%

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

      3. difference-of-squares 64.6%

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

   7. Applied egg-rr 79.3%

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

   8. Taylor expanded in y around 0 64.2%

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

      1. associate-/l* 69.0%

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

   10. Simplified 69.0%

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

   if 4.1999999999999997e100 < y

   1. Initial program 26.4%

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

      1. remove-double-neg 26.4%

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

      2. distribute-lft-neg-out 26.4%

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

      3. distribute-frac-neg2 26.4%

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

      4. distribute-frac-neg 26.4%

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

      5. neg-mul-1 26.4%

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

      6. distribute-lft-neg-out 26.4%

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

      7. *-commutative 26.4%

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

      8. distribute-lft-neg-in 26.4%

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

      9. times-frac 26.4%

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

      10. metadata-eval 26.4%

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

      11. metadata-eval 26.4%

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

      12. associate--l+ 26.4%

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

      13. fma-define 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in y around inf 84.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\frac{x - z}{y} \cdot \left(x + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot \left(x - z\_m\right)\right) \cdot \frac{0.5}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= y_m 1.25e+77) (* (* x (- x z_m)) (/ 0.5 y_m)) (* y_m 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (y_m <= 1.25e+77) {
		tmp = (x * (x - z_m)) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y_m <= 1.25d+77) then
        tmp = (x * (x - z_m)) * (0.5d0 / y_m)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (y_m <= 1.25e+77) {
		tmp = (x * (x - z_m)) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if y_m <= 1.25e+77:
		tmp = (x * (x - z_m)) * (0.5 / y_m)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (y_m <= 1.25e+77)
		tmp = Float64(Float64(x * Float64(x - z_m)) * Float64(0.5 / y_m));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 1.25e+77)
		tmp = (x * (x - z_m)) * (0.5 / y_m);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 1.25e+77], N[(N[(x * N[(x - z$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000001e77

   1. Initial program 72.7%

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

      1. remove-double-neg 72.7%

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

      2. distribute-lft-neg-out 72.7%

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

      3. distribute-frac-neg2 72.7%

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

      4. distribute-frac-neg 72.7%

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

      5. neg-mul-1 72.7%

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

      6. distribute-lft-neg-out 72.7%

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

      7. *-commutative 72.7%

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

      8. distribute-lft-neg-in 72.7%

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

      9. times-frac 72.3%

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

      10. metadata-eval 72.3%

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

      11. metadata-eval 72.3%

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

      12. associate--l+ 72.3%

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

      13. fma-define 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y around 0 60.6%

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

      1. associate-*r/ 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-/l* 61.0%

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

   7. Simplified 61.0%

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

      1. pow2 61.0%

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

      2. pow2 61.0%

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

      3. difference-of-squares 65.6%

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

   9. Applied egg-rr 65.6%

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

   10. Taylor expanded in x around inf 42.0%

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

   if 1.25000000000000001e77 < y

   1. Initial program 34.9%

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

      1. remove-double-neg 34.9%

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

      2. distribute-lft-neg-out 34.9%

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

      3. distribute-frac-neg2 34.9%

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

      4. distribute-frac-neg 34.9%

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

      5. neg-mul-1 34.9%

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

      6. distribute-lft-neg-out 34.9%

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

      7. *-commutative 34.9%

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

      8. distribute-lft-neg-in 34.9%

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

      9. times-frac 34.9%

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

      10. metadata-eval 34.9%

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

      11. metadata-eval 34.9%

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

      12. associate--l+ 34.9%

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

      13. fma-define 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in y around inf 77.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;\left(x \cdot \left(x - z\right)\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(-0.5 \cdot \frac{x + z\_m}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= z_m 2.1e+63) (* y_m 0.5) (* z_m (* -0.5 (/ (+ x z_m) y_m))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.1e+63) {
		tmp = y_m * 0.5;
	} else {
		tmp = z_m * (-0.5 * ((x + z_m) / y_m));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.1d+63) then
        tmp = y_m * 0.5d0
    else
        tmp = z_m * ((-0.5d0) * ((x + z_m) / y_m))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (z_m <= 2.1e+63) {
		tmp = y_m * 0.5;
	} else {
		tmp = z_m * (-0.5 * ((x + z_m) / y_m));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 2.1e+63:
		tmp = y_m * 0.5
	else:
		tmp = z_m * (-0.5 * ((x + z_m) / y_m))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 2.1e+63)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(z_m * Float64(-0.5 * Float64(Float64(x + z_m) / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 2.1e+63)
		tmp = y_m * 0.5;
	else
		tmp = z_m * (-0.5 * ((x + z_m) / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 2.1e+63], N[(y$95$m * 0.5), $MachinePrecision], N[(z$95$m * N[(-0.5 * N[(N[(x + z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.1000000000000002e63

   1. Initial program 67.6%

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

      1. remove-double-neg 67.6%

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

      2. distribute-lft-neg-out 67.6%

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

      3. distribute-frac-neg2 67.6%

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

      4. distribute-frac-neg 67.6%

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

      5. neg-mul-1 67.6%

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

      6. distribute-lft-neg-out 67.6%

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

      7. *-commutative 67.6%

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

      8. distribute-lft-neg-in 67.6%

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

      9. times-frac 67.2%

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

      10. metadata-eval 67.2%

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

      11. metadata-eval 67.2%

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

      12. associate--l+ 67.2%

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

      13. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 44.6%

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

   if 2.1000000000000002e63 < z

   1. Initial program 63.6%

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

      1. remove-double-neg 63.6%

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

      2. distribute-lft-neg-out 63.6%

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

      3. distribute-frac-neg2 63.6%

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

      4. distribute-frac-neg 63.6%

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

      5. neg-mul-1 63.6%

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

      6. distribute-lft-neg-out 63.6%

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

      7. *-commutative 63.6%

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

      8. distribute-lft-neg-in 63.6%

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

      9. times-frac 63.6%

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

      10. metadata-eval 63.6%

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

      11. metadata-eval 63.6%

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

      12. associate--l+ 63.6%

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

      13. fma-define 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in y around 0 66.0%

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

      1. associate-*r/ 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-/l* 66.0%

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

   7. Simplified 66.0%

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

      1. pow2 66.0%

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

      2. pow2 66.0%

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

      3. difference-of-squares 80.2%

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

   9. Applied egg-rr 80.2%

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

   10. Taylor expanded in x around 0 70.7%

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

       1. neg-mul-1 70.7%

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

   12. Simplified 70.7%

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

   13. Taylor expanded in z around 0 65.4%

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

       1. distribute-lft-out 65.4%

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

       2. *-lft-identity 65.4%

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

       3. associate-*l/ 65.4%

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

       4. *-lft-identity 65.4%

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

       5. associate-*l/ 65.4%

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

       6. distribute-lft-in 74.7%

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

       7. *-commutative 74.7%

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

       8. associate-*r/ 74.7%

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

       9. *-rgt-identity 74.7%

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

       10. +-commutative 74.7%

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

   15. Simplified 74.7%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+63}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{x + z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.2 \cdot 10^{+63}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot z\_m\right) \cdot \frac{-0.5}{y\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= z_m 5.2e+63) (* y_m 0.5) (* (* z_m z_m) (/ (- 0.5) y_m)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 5.2e+63) {
		tmp = y_m * 0.5;
	} else {
		tmp = (z_m * z_m) * (-0.5 / y_m);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.2d+63) then
        tmp = y_m * 0.5d0
    else
        tmp = (z_m * z_m) * (-0.5d0 / y_m)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (z_m <= 5.2e+63) {
		tmp = y_m * 0.5;
	} else {
		tmp = (z_m * z_m) * (-0.5 / y_m);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 5.2e+63:
		tmp = y_m * 0.5
	else:
		tmp = (z_m * z_m) * (-0.5 / y_m)
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 5.2e+63)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(Float64(z_m * z_m) * Float64(Float64(-0.5) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 5.2e+63)
		tmp = y_m * 0.5;
	else
		tmp = (z_m * z_m) * (-0.5 / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 5.2e+63], N[(y$95$m * 0.5), $MachinePrecision], N[(N[(z$95$m * z$95$m), $MachinePrecision] * N[((-0.5) / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.2000000000000002e63

   1. Initial program 67.6%

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

      1. remove-double-neg 67.6%

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

      2. distribute-lft-neg-out 67.6%

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

      3. distribute-frac-neg2 67.6%

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

      4. distribute-frac-neg 67.6%

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

      5. neg-mul-1 67.6%

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

      6. distribute-lft-neg-out 67.6%

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

      7. *-commutative 67.6%

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

      8. distribute-lft-neg-in 67.6%

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

      9. times-frac 67.2%

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

      10. metadata-eval 67.2%

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

      11. metadata-eval 67.2%

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

      12. associate--l+ 67.2%

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

      13. fma-define 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 44.6%

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

   if 5.2000000000000002e63 < z

   1. Initial program 63.6%

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

      1. remove-double-neg 63.6%

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

      2. distribute-lft-neg-out 63.6%

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

      3. distribute-frac-neg2 63.6%

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

      4. distribute-frac-neg 63.6%

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

      5. neg-mul-1 63.6%

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

      6. distribute-lft-neg-out 63.6%

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

      7. *-commutative 63.6%

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

      8. distribute-lft-neg-in 63.6%

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

      9. times-frac 63.6%

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

      10. metadata-eval 63.6%

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

      11. metadata-eval 63.6%

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

      12. associate--l+ 63.6%

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

      13. fma-define 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in y around 0 66.0%

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

      1. associate-*r/ 66.0%

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

      2. *-commutative 66.0%

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

      3. associate-/l* 66.0%

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

   7. Simplified 66.0%

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

      1. pow2 66.0%

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

      2. pow2 66.0%

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

      3. difference-of-squares 80.2%

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

   9. Applied egg-rr 80.2%

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

   10. Taylor expanded in x around 0 70.7%

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

       1. neg-mul-1 70.7%

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

   12. Simplified 70.7%

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

   13. Taylor expanded in x around 0 65.6%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+63}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{-0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \frac{z\_m}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= z_m 7.8e+154) (* y_m 0.5) (* -0.5 (* x (/ z_m y_m))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (z_m <= 7.8e+154) {
		tmp = y_m * 0.5;
	} else {
		tmp = -0.5 * (x * (z_m / y_m));
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 7.8d+154) then
        tmp = y_m * 0.5d0
    else
        tmp = (-0.5d0) * (x * (z_m / y_m))
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
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_m) {
	double tmp;
	if (z_m <= 7.8e+154) {
		tmp = y_m * 0.5;
	} else {
		tmp = -0.5 * (x * (z_m / y_m));
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if z_m <= 7.8e+154:
		tmp = y_m * 0.5
	else:
		tmp = -0.5 * (x * (z_m / y_m))
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (z_m <= 7.8e+154)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(-0.5 * Float64(x * Float64(z_m / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (z_m <= 7.8e+154)
		tmp = y_m * 0.5;
	else
		tmp = -0.5 * (x * (z_m / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * If[LessEqual[z$95$m, 7.8e+154], N[(y$95$m * 0.5), $MachinePrecision], N[(-0.5 * N[(x * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.8000000000000006e154

   1. Initial program 68.3%

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

      1. remove-double-neg 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-frac-neg2 68.3%

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

      4. distribute-frac-neg 68.3%

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

      5. neg-mul-1 68.3%

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

      6. distribute-lft-neg-out 68.3%

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

      7. *-commutative 68.3%

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

      8. distribute-lft-neg-in 68.3%

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

      9. times-frac 68.0%

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

      10. metadata-eval 68.0%

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

      11. metadata-eval 68.0%

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

      12. associate--l+ 68.0%

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

      13. fma-define 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in y around inf 42.2%

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

   if 7.8000000000000006e154 < z

   1. Initial program 52.9%

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

      1. remove-double-neg 52.9%

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

      2. distribute-lft-neg-out 52.9%

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

      3. distribute-frac-neg2 52.9%

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

      4. distribute-frac-neg 52.9%

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

      5. neg-mul-1 52.9%

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

      6. distribute-lft-neg-out 52.9%

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

      7. *-commutative 52.9%

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

      8. distribute-lft-neg-in 52.9%

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

      9. times-frac 52.9%

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

      10. metadata-eval 52.9%

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

      11. metadata-eval 52.9%

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

      12. associate--l+ 52.9%

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

      13. fma-define 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-*r/ 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-/l* 57.5%

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

   7. Simplified 57.5%

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

      1. pow2 57.5%

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

      2. pow2 57.5%

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

      3. difference-of-squares 83.6%

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

   9. Applied egg-rr 83.6%

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

   10. Taylor expanded in x around 0 83.6%

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

       1. neg-mul-1 83.6%

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

   12. Simplified 83.6%

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

   13. Taylor expanded in x around inf 23.4%

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

       1. associate-/l* 27.3%

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

   15. Simplified 27.3%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

z_m = Math.abs(z);
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_m) {
	return y_s * (y_m * 0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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$95$m_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.9%

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

   1. remove-double-neg 66.9%

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

   2. distribute-lft-neg-out 66.9%

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

   3. distribute-frac-neg2 66.9%

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

   4. distribute-frac-neg 66.9%

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

   5. neg-mul-1 66.9%

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

   6. distribute-lft-neg-out 66.9%

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

   7. *-commutative 66.9%

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

   8. distribute-lft-neg-in 66.9%

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

   9. times-frac 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 68.2%

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

3. Simplified 68.2%

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

5. Taylor expanded in y around inf 38.9%

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

6. Final simplification 38.9%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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