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

Percentage Accurate: 69.9% → 96.0%

Time: 10.8s

Alternatives: 6

Speedup: 1.1×

• 42.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.9% 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: 96.0% accurate, 0.1× speedup

Math

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

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 x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
      (* 0.5 (* (+ y_m z_m) (/ (- y_m z_m) y_m)))
      (* 0.5 (/ (hypot x y_m) (/ y_m (hypot x 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	} else {
		tmp = 0.5 * (hypot(x, y_m) / (y_m / hypot(x, 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	} else {
		tmp = 0.5 * (Math.hypot(x, y_m) / (y_m / Math.hypot(x, 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= math.inf):
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m))
	else:
		tmp = 0.5 * (math.hypot(x, y_m) / (y_m / math.hypot(x, 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(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= Inf))
		tmp = Float64(0.5 * Float64(Float64(y_m + z_m) * Float64(Float64(y_m - z_m) / y_m)));
	else
		tmp = Float64(0.5 * Float64(hypot(x, y_m) / Float64(y_m / hypot(x, 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= Inf)))
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	else
		tmp = 0.5 * (hypot(x, y_m) / (y_m / hypot(x, 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[(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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(y$95$m / N[Sqrt[x ^ 2 + y$95$m ^ 2], $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))) < 0.0 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.5%

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

      1. remove-double-neg 61.5%

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

      2. distribute-lft-neg-out 61.5%

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

      3. distribute-frac-neg2 61.5%

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

      4. distribute-frac-neg 61.5%

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

      5. neg-mul-1 61.5%

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

      6. distribute-lft-neg-out 61.5%

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

      7. *-commutative 61.5%

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

      8. distribute-lft-neg-in 61.5%

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

      9. times-frac 61.5%

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

      10. metadata-eval 61.5%

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

      11. metadata-eval 61.5%

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

      12. associate--l+ 61.5%

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

      13. fma-define 66.7%

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

   3. Simplified 66.7%

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

      1. difference-of-squares 70.2%

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

      2. *-commutative 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0 48.8%

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

      1. associate-/l* 71.2%

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

   9. Simplified 71.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.1%

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

      1. remove-double-neg 75.1%

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

      2. distribute-lft-neg-out 75.1%

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

      3. distribute-frac-neg2 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. neg-mul-1 75.1%

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

      6. distribute-lft-neg-out 75.1%

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

      7. *-commutative 75.1%

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

      8. distribute-lft-neg-in 75.1%

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

      9. times-frac 75.1%

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

      10. metadata-eval 75.1%

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

      11. metadata-eval 75.1%

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

      12. associate--l+ 75.1%

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

      13. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in z around 0 46.7%

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

      1. +-commutative 46.7%

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

      2. unpow2 46.7%

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

      3. fma-undefine 46.7%

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

   7. Simplified 46.7%

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

      1. fma-undefine 46.7%

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

      2. pow2 46.7%

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

      3. +-commutative 46.7%

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

      4. add-sqr-sqrt 46.4%

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

      5. sqrt-div 46.4%

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

      6. hypot-define 46.4%

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

      7. sqrt-div 46.3%

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

      8. hypot-define 69.3%

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

   9. Applied egg-rr 69.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y}} \cdot \frac{\mathsf{hypot}\left(x, y\right)}{\sqrt{y}}\right)} \]
   10. Step-by-step derivation

       1. unpow2 69.3%

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

       2. hypot-undefine 46.3%

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

       3. unpow2 46.3%

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

       4. unpow2 46.3%

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

       5. +-commutative 46.3%

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

       6. unpow2 46.3%

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

       7. unpow2 46.3%

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

       8. hypot-define 69.3%

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

   11. Simplified 69.3%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{\mathsf{hypot}\left(y, x\right)}{\sqrt{y}}\right)}^{2}} \]
   12. Step-by-step derivation

       1. unpow2 69.3%

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

       2. clear-num 69.2%

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

       3. frac-times 69.3%

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

       4. *-un-lft-identity 69.3%

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

   13. Applied egg-rr 69.3%

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

       1. hypot-undefine 46.3%

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

       2. unpow2 46.3%

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

       3. unpow2 46.3%

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

       4. +-commutative 46.3%

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

       5. unpow2 46.3%

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

       6. unpow2 46.3%

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

       7. hypot-undefine 69.3%

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

       8. associate-*l/ 69.3%

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

       9. rem-square-sqrt 69.9%

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

       10. hypot-undefine 46.7%

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

       11. unpow2 46.7%

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

       12. unpow2 46.7%

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

       13. +-commutative 46.7%

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

       14. unpow2 46.7%

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

       15. unpow2 46.7%

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

       16. hypot-undefine 69.9%

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

   15. Simplified 69.9%

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

4. Final simplification 70.6%

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

Alternative 2: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, \left(y\_m + z\_m\right) \cdot \left(y\_m - z\_m\right)\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + z\_m\right) \cdot \frac{y\_m - z\_m}{y\_m}\right)\\ \end{array} \end{array} \]

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 (<= y_m 3.7e+112)
    (* 0.5 (/ (fma x x (* (+ y_m z_m) (- y_m z_m))) y_m))
    (* 0.5 (* (+ y_m 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 (y_m <= 3.7e+112) {
		tmp = 0.5 * (fma(x, x, ((y_m + z_m) * (y_m - z_m))) / y_m);
	} else {
		tmp = 0.5 * ((y_m + 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 (y_m <= 3.7e+112)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y_m + z_m) * Float64(y_m - z_m))) / y_m));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + z_m) * Float64(Float64(y_m - z_m) / y_m)));
	end
	return Float64(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, 3.7e+112], N[(0.5 * N[(N[(x * x + N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.70000000000000004e112

   1. Initial program 79.3%

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

      1. remove-double-neg 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-frac-neg2 79.3%

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

      4. distribute-frac-neg 79.3%

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

      5. neg-mul-1 79.3%

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

      6. distribute-lft-neg-out 79.3%

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

      7. *-commutative 79.3%

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

      8. distribute-lft-neg-in 79.3%

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

      9. times-frac 79.3%

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

      10. metadata-eval 79.3%

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

      11. metadata-eval 79.3%

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

      12. associate--l+ 79.3%

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

      13. fma-define 82.8%

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

   3. Simplified 82.8%

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

      1. difference-of-squares 83.9%

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

      2. *-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

   if 3.70000000000000004e112 < y

   1. Initial program 26.2%

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

      1. remove-double-neg 26.2%

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

      2. distribute-lft-neg-out 26.2%

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

      3. distribute-frac-neg2 26.2%

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

      4. distribute-frac-neg 26.2%

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

      5. neg-mul-1 26.2%

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

      6. distribute-lft-neg-out 26.2%

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

      7. *-commutative 26.2%

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

      8. distribute-lft-neg-in 26.2%

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

      9. times-frac 26.2%

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

      10. metadata-eval 26.2%

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

      11. metadata-eval 26.2%

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

      12. associate--l+ 26.2%

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

      13. fma-define 26.2%

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

   3. Simplified 26.2%

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

      1. difference-of-squares 30.7%

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

      2. *-commutative 30.7%

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

   6. Applied egg-rr 30.7%

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

   7. Taylor expanded in x around 0 29.0%

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

      1. associate-/l* 87.8%

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

   9. Simplified 87.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.3× speedup

Math

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

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 x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
      (* 0.5 (* (+ y_m z_m) (/ (- y_m z_m) y_m)))
      (* 0.5 (+ y_m (/ (* x x) 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	} else {
		tmp = 0.5 * (y_m + ((x * x) / 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	} else {
		tmp = 0.5 * (y_m + ((x * x) / 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= math.inf):
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m))
	else:
		tmp = 0.5 * (y_m + ((x * x) / 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(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= Inf))
		tmp = Float64(0.5 * Float64(Float64(y_m + z_m) * Float64(Float64(y_m - z_m) / y_m)));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(Float64(x * x) / 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 * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= Inf)))
		tmp = 0.5 * ((y_m + z_m) * ((y_m - z_m) / y_m));
	else
		tmp = 0.5 * (y_m + ((x * x) / 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[(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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(N[(x * x), $MachinePrecision] / y$95$m), $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))) < 0.0 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.5%

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

      1. remove-double-neg 61.5%

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

      2. distribute-lft-neg-out 61.5%

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

      3. distribute-frac-neg2 61.5%

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

      4. distribute-frac-neg 61.5%

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

      5. neg-mul-1 61.5%

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

      6. distribute-lft-neg-out 61.5%

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

      7. *-commutative 61.5%

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

      8. distribute-lft-neg-in 61.5%

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

      9. times-frac 61.5%

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

      10. metadata-eval 61.5%

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

      11. metadata-eval 61.5%

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

      12. associate--l+ 61.5%

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

      13. fma-define 66.7%

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

   3. Simplified 66.7%

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

      1. difference-of-squares 70.2%

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

      2. *-commutative 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in x around 0 48.8%

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

      1. associate-/l* 71.2%

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

   9. Simplified 71.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.1%

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

      1. remove-double-neg 75.1%

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

      2. distribute-lft-neg-out 75.1%

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

      3. distribute-frac-neg2 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. neg-mul-1 75.1%

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

      6. distribute-lft-neg-out 75.1%

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

      7. *-commutative 75.1%

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

      8. distribute-lft-neg-in 75.1%

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

      9. times-frac 75.1%

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

      10. metadata-eval 75.1%

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

      11. metadata-eval 75.1%

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

      12. associate--l+ 75.1%

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

      13. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in z around 0 46.7%

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

      1. +-commutative 46.7%

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

      2. unpow2 46.7%

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

      3. fma-undefine 46.7%

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

   7. Simplified 46.7%

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

   8. Taylor expanded in x around 0 62.3%

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

      1. distribute-lft-out 62.3%

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

   10. Simplified 62.3%

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

       1. pow2 62.3%

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

   12. Applied egg-rr 62.3%

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

4. Final simplification 67.0%

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

Alternative 4: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{+141}:\\ \;\;\;\;0.5 \cdot \left(y\_m + \frac{x \cdot x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{y\_m - z\_m}{y\_m}\right)\\ \end{array} \end{array} \]

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+141)
    (* 0.5 (+ y_m (/ (* x x) y_m)))
    (* 0.5 (* 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 <= 2.1e+141) {
		tmp = 0.5 * (y_m + ((x * x) / y_m));
	} else {
		tmp = 0.5 * (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 <= 2.1d+141) then
        tmp = 0.5d0 * (y_m + ((x * x) / y_m))
    else
        tmp = 0.5d0 * (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 <= 2.1e+141) {
		tmp = 0.5 * (y_m + ((x * x) / y_m));
	} else {
		tmp = 0.5 * (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 <= 2.1e+141:
		tmp = 0.5 * (y_m + ((x * x) / y_m))
	else:
		tmp = 0.5 * (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 <= 2.1e+141)
		tmp = Float64(0.5 * Float64(y_m + Float64(Float64(x * x) / y_m)));
	else
		tmp = Float64(0.5 * Float64(z_m * Float64(Float64(y_m - 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+141)
		tmp = 0.5 * (y_m + ((x * x) / y_m));
	else
		tmp = 0.5 * (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, 2.1e+141], N[(0.5 * N[(y$95$m + N[(N[(x * x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(z$95$m * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0999999999999998e141

   1. Initial program 74.0%

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

      1. remove-double-neg 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-frac-neg2 74.0%

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

      4. distribute-frac-neg 74.0%

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

      5. neg-mul-1 74.0%

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

      6. distribute-lft-neg-out 74.0%

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

      7. *-commutative 74.0%

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

      8. distribute-lft-neg-in 74.0%

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

      9. times-frac 74.0%

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

      10. metadata-eval 74.0%

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

      11. metadata-eval 74.0%

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

      12. associate--l+ 74.0%

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

      13. fma-define 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in z around 0 49.8%

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

      1. +-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. fma-undefine 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in x around 0 65.8%

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

      1. distribute-lft-out 65.8%

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

   10. Simplified 65.8%

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

       1. pow2 65.8%

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

   12. Applied egg-rr 65.8%

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

   if 2.0999999999999998e141 < z

   1. Initial program 35.4%

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

      1. remove-double-neg 35.4%

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

      2. distribute-lft-neg-out 35.4%

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

      3. distribute-frac-neg2 35.4%

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

      4. distribute-frac-neg 35.4%

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

      5. neg-mul-1 35.4%

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

      6. distribute-lft-neg-out 35.4%

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

      7. *-commutative 35.4%

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

      8. distribute-lft-neg-in 35.4%

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

      9. times-frac 35.4%

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

      10. metadata-eval 35.4%

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

      11. metadata-eval 35.4%

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

      12. associate--l+ 35.4%

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

      13. fma-define 45.3%

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

   3. Simplified 45.3%

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

      1. difference-of-squares 56.4%

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

      2. *-commutative 56.4%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in x around 0 56.4%

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

      1. associate-/l* 85.5%

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

   9. Simplified 85.5%

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

   10. Taylor expanded in y around 0 69.3%

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

Alternative 5: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 9.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{y\_m - z\_m}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

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 9.2e+113) (* 0.5 (* z_m (/ (- y_m z_m) 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 <= 9.2e+113) {
		tmp = 0.5 * (z_m * ((y_m - z_m) / 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 <= 9.2d+113) then
        tmp = 0.5d0 * (z_m * ((y_m - z_m) / 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 <= 9.2e+113) {
		tmp = 0.5 * (z_m * ((y_m - z_m) / 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 <= 9.2e+113:
		tmp = 0.5 * (z_m * ((y_m - z_m) / 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 <= 9.2e+113)
		tmp = Float64(0.5 * Float64(z_m * Float64(Float64(y_m - z_m) / 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 <= 9.2e+113)
		tmp = 0.5 * (z_m * ((y_m - z_m) / 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, 9.2e+113], N[(0.5 * N[(z$95$m * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.19999999999999987e113

   1. Initial program 78.6%

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

      1. remove-double-neg 78.6%

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

      2. distribute-lft-neg-out 78.6%

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

      3. distribute-frac-neg2 78.6%

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

      4. distribute-frac-neg 78.6%

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

      5. neg-mul-1 78.6%

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

      6. distribute-lft-neg-out 78.6%

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

      7. *-commutative 78.6%

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

      8. distribute-lft-neg-in 78.6%

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

      9. times-frac 78.6%

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

      10. metadata-eval 78.6%

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

      11. metadata-eval 78.6%

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

      12. associate--l+ 78.6%

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

      13. fma-define 82.0%

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

   3. Simplified 82.0%

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

      1. difference-of-squares 83.2%

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

      2. *-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around 0 55.8%

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

      1. associate-/l* 67.6%

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

   9. Simplified 67.6%

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

   10. Taylor expanded in y around 0 42.0%

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

   if 9.19999999999999987e113 < y

   1. Initial program 26.8%

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

      1. remove-double-neg 26.8%

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

      2. distribute-lft-neg-out 26.8%

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

      3. distribute-frac-neg2 26.8%

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

      4. distribute-frac-neg 26.8%

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

      5. neg-mul-1 26.8%

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

      6. distribute-lft-neg-out 26.8%

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

      7. *-commutative 26.8%

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

      8. distribute-lft-neg-in 26.8%

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

      9. times-frac 26.8%

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

      10. metadata-eval 26.8%

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

      11. metadata-eval 26.8%

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

      12. associate--l+ 26.8%

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

      13. fma-define 26.8%

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

   3. Simplified 26.8%

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

   5. Taylor expanded in y around inf 74.9%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.4% 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 67.9%

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

   1. remove-double-neg 67.9%

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

   2. distribute-lft-neg-out 67.9%

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

   3. distribute-frac-neg2 67.9%

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

   4. distribute-frac-neg 67.9%

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

   5. neg-mul-1 67.9%

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

   6. distribute-lft-neg-out 67.9%

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

   7. *-commutative 67.9%

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

   8. distribute-lft-neg-in 67.9%

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

   9. times-frac 67.9%

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

   10. metadata-eval 67.9%

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

   11. metadata-eval 67.9%

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

   12. associate--l+ 67.9%

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

   13. fma-define 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in y around inf 37.0%

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

6. Final simplification 37.0%

   \[\leadsto y \cdot 0.5 \]
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 2024160 
(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)))