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

Percentage Accurate: 69.2% → 96.0%

Time: 17.4s

Alternatives: 10

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% 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, 0.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\frac{y\_m + z}{\frac{y\_m \cdot 2}{y\_m - z}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\_m\right), z\right) \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y\_m, x\right)}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -4e-47)
      (/ (+ y_m z) (/ (* y_m 2.0) (- y_m z)))
      (if (<= t_0 INFINITY)
        (* (hypot (hypot x y_m) z) (* 0.5 (/ (hypot y_m x) y_m)))
        (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -4e-47) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = hypot(hypot(x, y_m), z) * (0.5 * (hypot(y_m, x) / y_m));
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -4e-47) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.hypot(Math.hypot(x, y_m), z) * (0.5 * (Math.hypot(y_m, x) / y_m));
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -4e-47:
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z))
	elif t_0 <= math.inf:
		tmp = math.hypot(math.hypot(x, y_m), z) * (0.5 * (math.hypot(y_m, x) / y_m))
	else:
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-47)
		tmp = Float64(Float64(y_m + z) / Float64(Float64(y_m * 2.0) / Float64(y_m - z)));
	elseif (t_0 <= Inf)
		tmp = Float64(hypot(hypot(x, y_m), z) * Float64(0.5 * Float64(hypot(y_m, x) / y_m)));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-47)
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	elseif (t_0 <= Inf)
		tmp = hypot(hypot(x, y_m), z) * (0.5 * (hypot(y_m, x) / y_m));
	else
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -4e-47], N[(N[(y$95$m + z), $MachinePrecision] / N[(N[(y$95$m * 2.0), $MachinePrecision] / N[(y$95$m - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] ^ 2 + z ^ 2], $MachinePrecision] * N[(0.5 * N[(N[Sqrt[y$95$m ^ 2 + x ^ 2], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y 2)) < -3.9999999999999999e-47

   1. Initial program 80.7%

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

      1. associate--l+ 80.7%

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

      2. +-commutative 80.7%

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

      3. sqr-neg 80.7%

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

      4. difference-of-squares 80.7%

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

      5. fma-def 80.7%

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

      6. sub-neg 80.7%

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

      7. sub-neg 80.7%

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

      8. remove-double-neg 80.7%

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

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.2%

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

      1. associate-/l* 67.0%

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

      2. div-inv 67.0%

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

      3. *-commutative 67.0%

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

      4. *-un-lft-identity 67.0%

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

      5. times-frac 67.0%

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

      6. metadata-eval 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-rgt-identity 67.0%

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

      3. associate-*r/ 67.0%

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

      4. *-commutative 67.0%

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

   9. Simplified 67.0%

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

   if -3.9999999999999999e-47 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y 2)) < +inf.0

   1. Initial program 74.2%

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

      1. expm1-log1p-u 71.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\right)\right)} \]

      2. expm1-udef 69.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\right)} - 1} \]

   4. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \frac{0.5}{y}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 71.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right) \cdot \frac{0.5}{y}\right)\right)} \]

      2. expm1-log1p 74.2%

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

      3. *-commutative 74.2%

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

   6. Simplified 74.2%

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

      1. add-cube-cbrt 73.6%

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

      2. pow3 73.6%

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

   8. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{0.5}{y} \cdot \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 74.2%

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

      2. *-commutative 74.2%

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

      3. add-sqr-sqrt 41.5%

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

      4. associate-*l* 41.5%

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

   10. Applied egg-rr 67.2%

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

   11. Taylor expanded in z around 0 41.7%

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

       1. associate-*l/ 41.7%

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

       2. *-lft-identity 41.7%

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

       3. +-commutative 41.7%

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

       4. unpow2 41.7%

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

       5. unpow2 41.7%

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

       6. hypot-def 67.3%

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

   13. Simplified 67.3%

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

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

   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. associate--l+ 0.0%

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

      2. +-commutative 0.0%

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

      3. sqr-neg 0.0%

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

      4. difference-of-squares 18.2%

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

      5. fma-def 45.8%

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

      6. sub-neg 45.8%

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

      7. sub-neg 45.8%

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

      8. remove-double-neg 45.8%

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

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 33.6%

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

      1. *-commutative 33.6%

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

      2. times-frac 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\frac{y + z}{\frac{y \cdot 2}{y - z}}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right) \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y, x\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.3e+156)
    (/ (- (fma x x (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.3e+156) {
		tmp = (fma(x, x, (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.3e+156)
		tmp = Float64(Float64(fma(x, x, Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	end
	return Float64(y_s * tmp)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.3e+156], N[(N[(N[(x * x + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e156

   1. Initial program 78.3%

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

      1. div-sub 74.2%

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

      2. sqr-neg 74.2%

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

      3. div-sub 78.3%

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

      4. sqr-neg 78.3%

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

      5. fma-def 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot y\right) - z \cdot z}{y \cdot 2}} \]
   4. Add Preprocessing

   if 3.2999999999999999e156 < y

   1. Initial program 9.7%

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

      1. associate--l+ 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. difference-of-squares 18.3%

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

      5. fma-def 20.8%

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

      6. sub-neg 20.8%

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

      7. sub-neg 20.8%

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

      8. remove-double-neg 20.8%

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

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 18.3%

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

      1. *-commutative 18.3%

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

      2. times-frac 86.1%

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m - z, y\_m + z, x \cdot x\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.3e+156)
    (/ (fma (- y_m z) (+ y_m z) (* x x)) (* y_m 2.0))
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.3e+156) {
		tmp = fma((y_m - z), (y_m + z), (x * x)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.3e+156)
		tmp = Float64(fma(Float64(y_m - z), Float64(y_m + z), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	end
	return Float64(y_s * tmp)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.3e+156], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e156

   1. Initial program 78.3%

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

      1. associate--l+ 78.3%

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

      2. +-commutative 78.3%

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

      3. sqr-neg 78.3%

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

      4. difference-of-squares 79.5%

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

      5. fma-def 83.3%

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

      6. sub-neg 83.3%

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

      7. sub-neg 83.3%

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

      8. remove-double-neg 83.3%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   if 3.2999999999999999e156 < y

   1. Initial program 9.7%

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

      1. associate--l+ 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. difference-of-squares 18.3%

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

      5. fma-def 20.8%

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

      6. sub-neg 20.8%

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

      7. sub-neg 20.8%

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

      8. remove-double-neg 20.8%

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

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 18.3%

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

      1. *-commutative 18.3%

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

      2. times-frac 86.1%

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.06 \cdot 10^{-11} \lor \neg \left(y\_m \leq 4.35 \cdot 10^{+43}\right) \land y\_m \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= y_m 2.06e-11) (and (not (<= y_m 4.35e+43)) (<= y_m 4.2e+82)))
    (* x (* x (/ 0.5 y_m)))
    (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m <= 2.06e-11) || (!(y_m <= 4.35e+43) && (y_m <= 4.2e+82))) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y_m <= 2.06d-11) .or. (.not. (y_m <= 4.35d+43)) .and. (y_m <= 4.2d+82)) then
        tmp = x * (x * (0.5d0 / y_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m <= 2.06e-11) || (!(y_m <= 4.35e+43) && (y_m <= 4.2e+82))) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (y_m <= 2.06e-11) or (not (y_m <= 4.35e+43) and (y_m <= 4.2e+82)):
		tmp = x * (x * (0.5 / y_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((y_m <= 2.06e-11) || (!(y_m <= 4.35e+43) && (y_m <= 4.2e+82)))
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((y_m <= 2.06e-11) || (~((y_m <= 4.35e+43)) && (y_m <= 4.2e+82)))
		tmp = x * (x * (0.5 / y_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[y$95$m, 2.06e-11], And[N[Not[LessEqual[y$95$m, 4.35e+43]], $MachinePrecision], LessEqual[y$95$m, 4.2e+82]]], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.0599999999999999e-11 or 4.34999999999999977e43 < y < 4.2e82

   1. Initial program 78.4%

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

   3. Taylor expanded in x around inf 36.1%

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

      1. div-inv 36.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{y \cdot 2}} \]

      2. metadata-eval 36.1%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 36.1%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 36.1%

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

      5. unpow2 36.1%

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

      6. associate-*l* 38.6%

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

   5. Applied egg-rr 38.6%

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

   if 2.0599999999999999e-11 < y < 4.34999999999999977e43 or 4.2e82 < y

   1. Initial program 39.1%

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

   3. Taylor expanded in y around inf 61.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.06 \cdot 10^{-11} \lor \neg \left(y \leq 4.35 \cdot 10^{+43}\right) \land y \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{-11} \lor \neg \left(y\_m \leq 4.2 \cdot 10^{+43}\right) \land y\_m \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= y_m 1.8e-11) (and (not (<= y_m 4.2e+43)) (<= y_m 1.8e+82)))
    (* (/ x y_m) (/ x 2.0))
    (* y_m 0.5))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m <= 1.8e-11) || (!(y_m <= 4.2e+43) && (y_m <= 1.8e+82))) {
		tmp = (x / y_m) * (x / 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y_m <= 1.8d-11) .or. (.not. (y_m <= 4.2d+43)) .and. (y_m <= 1.8d+82)) then
        tmp = (x / y_m) * (x / 2.0d0)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m <= 1.8e-11) || (!(y_m <= 4.2e+43) && (y_m <= 1.8e+82))) {
		tmp = (x / y_m) * (x / 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (y_m <= 1.8e-11) or (not (y_m <= 4.2e+43) and (y_m <= 1.8e+82)):
		tmp = (x / y_m) * (x / 2.0)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if ((y_m <= 1.8e-11) || (!(y_m <= 4.2e+43) && (y_m <= 1.8e+82)))
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((y_m <= 1.8e-11) || (~((y_m <= 4.2e+43)) && (y_m <= 1.8e+82)))
		tmp = (x / y_m) * (x / 2.0);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[Or[LessEqual[y$95$m, 1.8e-11], And[N[Not[LessEqual[y$95$m, 4.2e+43]], $MachinePrecision], LessEqual[y$95$m, 1.8e+82]]], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999992e-11 or 4.20000000000000003e43 < y < 1.80000000000000007e82

   1. Initial program 78.4%

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

   3. Taylor expanded in x around inf 36.1%

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

      1. unpow2 36.1%

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

      2. times-frac 38.6%

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

   5. Applied egg-rr 38.6%

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

   if 1.79999999999999992e-11 < y < 4.20000000000000003e43 or 1.80000000000000007e82 < y

   1. Initial program 39.1%

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

   3. Taylor expanded in y around inf 61.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-11} \lor \neg \left(y \leq 4.2 \cdot 10^{+43}\right) \land y \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y\_m}\\ \mathbf{elif}\;y\_m \leq 3.35 \cdot 10^{+44} \lor \neg \left(y\_m \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.25e-12)
    (/ (* x (* x 0.5)) y_m)
    (if (or (<= y_m 3.35e+44) (not (<= y_m 4.5e+82)))
      (* y_m 0.5)
      (* (/ x y_m) (/ x 2.0))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.25e-12) {
		tmp = (x * (x * 0.5)) / y_m;
	} else if ((y_m <= 3.35e+44) || !(y_m <= 4.5e+82)) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.25d-12) then
        tmp = (x * (x * 0.5d0)) / y_m
    else if ((y_m <= 3.35d+44) .or. (.not. (y_m <= 4.5d+82))) then
        tmp = y_m * 0.5d0
    else
        tmp = (x / y_m) * (x / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.25e-12) {
		tmp = (x * (x * 0.5)) / y_m;
	} else if ((y_m <= 3.35e+44) || !(y_m <= 4.5e+82)) {
		tmp = y_m * 0.5;
	} else {
		tmp = (x / y_m) * (x / 2.0);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.25e-12:
		tmp = (x * (x * 0.5)) / y_m
	elif (y_m <= 3.35e+44) or not (y_m <= 4.5e+82):
		tmp = y_m * 0.5
	else:
		tmp = (x / y_m) * (x / 2.0)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.25e-12)
		tmp = Float64(Float64(x * Float64(x * 0.5)) / y_m);
	elseif ((y_m <= 3.35e+44) || !(y_m <= 4.5e+82))
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(Float64(x / y_m) * Float64(x / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.25e-12)
		tmp = (x * (x * 0.5)) / y_m;
	elseif ((y_m <= 3.35e+44) || ~((y_m <= 4.5e+82)))
		tmp = y_m * 0.5;
	else
		tmp = (x / y_m) * (x / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.25e-12], N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], If[Or[LessEqual[y$95$m, 3.35e+44], N[Not[LessEqual[y$95$m, 4.5e+82]], $MachinePrecision]], N[(y$95$m * 0.5), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.24999999999999992e-12

   1. Initial program 78.0%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. div-inv 35.9%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{y \cdot 2}} \]

      2. metadata-eval 35.9%

         \[\leadsto {x}^{2} \cdot \frac{1}{y \cdot \color{blue}{\frac{1}{0.5}}} \]

      3. div-inv 35.9%

         \[\leadsto {x}^{2} \cdot \frac{1}{\color{blue}{\frac{y}{0.5}}} \]

      4. clear-num 35.9%

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

      5. unpow2 35.9%

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

      6. associate-*l* 38.0%

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

   5. Applied egg-rr 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-*r/ 38.0%

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

      3. associate-*l/ 35.9%

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

   7. Applied egg-rr 35.9%

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

   if 1.24999999999999992e-12 < y < 3.35000000000000018e44 or 4.4999999999999997e82 < y

   1. Initial program 39.1%

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

   3. Taylor expanded in y around inf 61.0%

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

   if 3.35000000000000018e44 < y < 4.4999999999999997e82

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf 40.0%

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

      1. unpow2 40.0%

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

      2. times-frac 51.4%

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

   5. Applied egg-rr 51.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{+44} \lor \neg \left(y \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 3.3e+156)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.3e+156) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 3.3d+156) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0d0)
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 3.3e+156) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 3.3e+156:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 3.3e+156)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 3.3e+156)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 3.3e+156], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e156

   1. Initial program 78.3%

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

   if 3.2999999999999999e156 < y

   1. Initial program 9.7%

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

      1. associate--l+ 9.7%

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

      2. +-commutative 9.7%

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

      3. sqr-neg 9.7%

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

      4. difference-of-squares 18.3%

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

      5. fma-def 20.8%

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

      6. sub-neg 20.8%

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

      7. sub-neg 20.8%

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

      8. remove-double-neg 20.8%

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

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 18.3%

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

      1. *-commutative 18.3%

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

      2. times-frac 86.1%

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

   7. Applied egg-rr 86.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 1.5e+141)
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0))
    (/ x (* y_m (/ 2.0 x))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.5e+141) {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.5d+141) then
        tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0d0)
    else
        tmp = x / (y_m * (2.0d0 / x))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 1.5e+141) {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 1.5e+141:
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	else:
		tmp = x / (y_m * (2.0 / x))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 1.5e+141)
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	else
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 1.5e+141)
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	else
		tmp = x / (y_m * (2.0 / x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 1.5e+141], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e141

   1. Initial program 71.0%

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

      1. associate--l+ 71.0%

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

      2. +-commutative 71.0%

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

      3. sqr-neg 71.0%

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

      4. difference-of-squares 73.3%

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

      5. fma-def 74.6%

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

      6. sub-neg 74.6%

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

      7. sub-neg 74.6%

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

      8. remove-double-neg 74.6%

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

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. times-frac 76.6%

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

   7. Applied egg-rr 76.6%

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

   if 1.4999999999999999e141 < x

   1. Initial program 46.5%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. unpow2 65.4%

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

      2. times-frac 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. clear-num 79.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{x}}} \cdot \frac{x}{y} \]

      3. frac-times 79.6%

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{2}{x} \cdot y}} \]

      4. *-un-lft-identity 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{y\_m + z}{\frac{y\_m \cdot 2}{y\_m - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y\_m \cdot \frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 8.6e+143)
    (/ (+ y_m z) (/ (* y_m 2.0) (- y_m z)))
    (/ x (* y_m (/ 2.0 x))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 8.6e+143) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 8.6d+143) then
        tmp = (y_m + z) / ((y_m * 2.0d0) / (y_m - z))
    else
        tmp = x / (y_m * (2.0d0 / x))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 8.6e+143) {
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	} else {
		tmp = x / (y_m * (2.0 / x));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 8.6e+143:
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z))
	else:
		tmp = x / (y_m * (2.0 / x))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 8.6e+143)
		tmp = Float64(Float64(y_m + z) / Float64(Float64(y_m * 2.0) / Float64(y_m - z)));
	else
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 8.6e+143)
		tmp = (y_m + z) / ((y_m * 2.0) / (y_m - z));
	else
		tmp = x / (y_m * (2.0 / x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 8.6e+143], N[(N[(y$95$m + z), $MachinePrecision] / N[(N[(y$95$m * 2.0), $MachinePrecision] / N[(y$95$m - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.60000000000000003e143

   1. Initial program 71.0%

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

      1. associate--l+ 71.0%

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

      2. +-commutative 71.0%

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

      3. sqr-neg 71.0%

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

      4. difference-of-squares 73.3%

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

      5. fma-def 74.6%

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

      6. sub-neg 74.6%

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

      7. sub-neg 74.6%

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

      8. remove-double-neg 74.6%

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

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.4%

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

      1. associate-/l* 76.6%

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

      2. div-inv 76.6%

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

      3. *-commutative 76.6%

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

      4. *-un-lft-identity 76.6%

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

      5. times-frac 76.6%

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

      6. metadata-eval 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-*r/ 76.6%

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

      2. *-rgt-identity 76.6%

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

      3. associate-*r/ 76.6%

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

      4. *-commutative 76.6%

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

   9. Simplified 76.6%

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

   if 8.60000000000000003e143 < x

   1. Initial program 46.5%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. unpow2 65.4%

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

      2. times-frac 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

      2. clear-num 79.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{x}}} \cdot \frac{x}{y} \]

      3. frac-times 79.6%

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{2}{x} \cdot y}} \]

      4. *-un-lft-identity 79.6%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{y + z}{\frac{y \cdot 2}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

3. Taylor expanded in y around inf 34.7%

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

4. Final simplification 34.7%

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

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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