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

Percentage Accurate: 69.2% → 93.3%

Time: 10.4s

Alternatives: 6

Speedup: 0.5×

• 41.7% 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: 93.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -4 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{hypot}\left(x, y\_m + z\_m\right) \cdot \frac{z\_m}{-y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

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 (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -4e-74)
    (* 0.5 (* (hypot x (+ y_m z_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 tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74) {
		tmp = 0.5 * (hypot(x, (y_m + z_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 tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74) {
		tmp = 0.5 * (Math.hypot(x, (y_m + z_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):
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74:
		tmp = 0.5 * (math.hypot(x, (y_m + z_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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -4e-74)
		tmp = Float64(0.5 * Float64(hypot(x, Float64(y_m + z_m)) * Float64(z_m / Float64(-y_m))));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(x * Float64(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)
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74)
		tmp = 0.5 * (hypot(x, (y_m + z_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_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -4e-74], N[(0.5 * N[(N[Sqrt[x ^ 2 + N[(y$95$m + z$95$m), $MachinePrecision] ^ 2], $MachinePrecision] * N[(z$95$m / (-y$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $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))) < -3.99999999999999983e-74

   1. Initial program 81.8%

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

      1. remove-double-neg 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-frac-neg 81.8%

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

      5. neg-mul-1 81.8%

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

      6. distribute-lft-neg-out 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-neg-in 81.8%

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

      9. times-frac 81.8%

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

      10. metadata-eval 81.8%

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

      11. metadata-eval 81.8%

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

      12. associate--l+ 81.8%

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

      13. fma-define 81.8%

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

   3. Simplified 81.8%

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

      1. prod-diff 68.7%

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

      2. fma-neg 68.7%

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

      3. difference-of-squares 68.7%

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

      4. fma-define 68.6%

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

      5. pow2 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. add-sqr-sqrt 54.1%

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

      2. associate-/l* 54.1%

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

   8. Applied egg-rr 70.2%

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

   9. Taylor expanded in z around -inf 25.5%

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

       1. associate-*r/ 25.5%

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

       2. mul-1-neg 25.5%

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

   11. Simplified 25.5%

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

   if -3.99999999999999983e-74 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.8%

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

      1. remove-double-neg 62.8%

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

      2. distribute-lft-neg-out 62.8%

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

      3. distribute-frac-neg2 62.8%

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

      4. distribute-frac-neg 62.8%

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

      5. neg-mul-1 62.8%

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

      6. distribute-lft-neg-out 62.8%

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

      7. *-commutative 62.8%

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

      8. distribute-lft-neg-in 62.8%

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

      9. times-frac 62.8%

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

      10. metadata-eval 62.8%

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

      11. metadata-eval 62.8%

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

      12. associate--l+ 62.8%

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

      13. fma-define 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in z around 0 41.4%

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

      1. associate-*r/ 41.4%

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

      2. rem-square-sqrt 41.4%

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

      3. unpow2 41.4%

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

      4. unpow2 41.4%

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

      5. hypot-undefine 41.4%

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

      6. unpow2 41.4%

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

      7. unpow2 41.4%

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

      8. hypot-undefine 41.4%

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

      9. unpow2 41.4%

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

      10. associate-*r/ 41.4%

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

      11. *-commutative 41.4%

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

      12. metadata-eval 41.4%

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

      13. times-frac 41.4%

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

      14. associate-/l* 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in x around 0 58.7%

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

      1. distribute-lft-out 58.7%

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

   10. Simplified 58.7%

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

       1. unpow2 58.7%

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

       2. associate-/l* 62.8%

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

   12. Applied egg-rr 62.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -4 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{hypot}\left(x, y + z\right) \cdot \frac{z}{-y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{\frac{y\_m \cdot -2}{z\_m}}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

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 (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -4e-74)
    (/ 1.0 (/ (/ (* y_m -2.0) z_m) z_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 tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74) {
		tmp = 1.0 / (((y_m * -2.0) / z_m) / z_m);
	} else {
		tmp = 0.5 * (y_m + (x * (x / 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 (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)) <= (-4d-74)) then
        tmp = 1.0d0 / (((y_m * (-2.0d0)) / z_m) / z_m)
    else
        tmp = 0.5d0 * (y_m + (x * (x / 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 (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74) {
		tmp = 1.0 / (((y_m * -2.0) / z_m) / z_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):
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74:
		tmp = 1.0 / (((y_m * -2.0) / z_m) / z_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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -4e-74)
		tmp = Float64(1.0 / Float64(Float64(Float64(y_m * -2.0) / z_m) / z_m));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64(x * Float64(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)
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -4e-74)
		tmp = 1.0 / (((y_m * -2.0) / z_m) / z_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_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -4e-74], N[(1.0 / N[(N[(N[(y$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $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))) < -3.99999999999999983e-74

   1. Initial program 81.8%

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

      1. remove-double-neg 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-frac-neg 81.8%

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

      5. neg-mul-1 81.8%

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

      6. distribute-lft-neg-out 81.8%

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

      7. *-commutative 81.8%

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

      8. distribute-lft-neg-in 81.8%

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

      9. times-frac 81.8%

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

      10. metadata-eval 81.8%

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

      11. metadata-eval 81.8%

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

      12. associate--l+ 81.8%

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

      13. fma-define 81.8%

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

   3. Simplified 81.8%

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

      1. fma-undefine 81.8%

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

      2. associate--l+ 81.8%

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

      3. metadata-eval 81.8%

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

      4. times-frac 81.8%

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

      5. *-un-lft-identity 81.8%

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

      6. *-commutative 81.8%

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

      7. clear-num 81.7%

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

      8. associate-/l* 81.6%

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

      9. add-sqr-sqrt 81.6%

         \[\leadsto \frac{1}{y \cdot \frac{2}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}} - z \cdot z}} \]

      10. pow2 81.6%

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

      11. hypot-define 81.6%

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

      12. pow2 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in z around inf 28.9%

      \[\leadsto \frac{1}{y \cdot \color{blue}{\frac{-2}{{z}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 28.9%

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

      2. unpow2 28.9%

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

      3. associate-/r* 30.4%

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

   9. Applied egg-rr 30.4%

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

   if -3.99999999999999983e-74 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.8%

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

      1. remove-double-neg 62.8%

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

      2. distribute-lft-neg-out 62.8%

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

      3. distribute-frac-neg2 62.8%

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

      4. distribute-frac-neg 62.8%

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

      5. neg-mul-1 62.8%

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

      6. distribute-lft-neg-out 62.8%

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

      7. *-commutative 62.8%

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

      8. distribute-lft-neg-in 62.8%

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

      9. times-frac 62.8%

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

      10. metadata-eval 62.8%

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

      11. metadata-eval 62.8%

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

      12. associate--l+ 62.8%

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

      13. fma-define 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in z around 0 41.4%

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

      1. associate-*r/ 41.4%

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

      2. rem-square-sqrt 41.4%

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

      3. unpow2 41.4%

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

      4. unpow2 41.4%

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

      5. hypot-undefine 41.4%

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

      6. unpow2 41.4%

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

      7. unpow2 41.4%

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

      8. hypot-undefine 41.4%

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

      9. unpow2 41.4%

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

      10. associate-*r/ 41.4%

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

      11. *-commutative 41.4%

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

      12. metadata-eval 41.4%

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

      13. times-frac 41.4%

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

      14. associate-/l* 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in x around 0 58.7%

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

      1. distribute-lft-out 58.7%

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

   10. Simplified 58.7%

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

       1. unpow2 58.7%

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

       2. associate-/l* 62.8%

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

   12. Applied egg-rr 62.8%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{\frac{y \cdot -2}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.45 \cdot 10^{+165}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{z\_m}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (* y_s (if (<= x 3.45e+165) (* y_m 0.5) (* 0.5 (* z_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 (x <= 3.45e+165) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * (z_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 (x <= 3.45d+165) then
        tmp = y_m * 0.5d0
    else
        tmp = 0.5d0 * (z_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 (x <= 3.45e+165) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * (z_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 x <= 3.45e+165:
		tmp = y_m * 0.5
	else:
		tmp = 0.5 * (z_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 (x <= 3.45e+165)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(0.5 * Float64(z_m * Float64(z_m / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (x <= 3.45e+165)
		tmp = y_m * 0.5;
	else
		tmp = 0.5 * (z_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[x, 3.45e+165], N[(y$95$m * 0.5), $MachinePrecision], N[(0.5 * N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.45000000000000003e165

   1. Initial program 73.0%

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

      1. remove-double-neg 73.0%

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

      2. distribute-lft-neg-out 73.0%

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

      3. distribute-frac-neg2 73.0%

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

      4. distribute-frac-neg 73.0%

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

      5. neg-mul-1 73.0%

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

      6. distribute-lft-neg-out 73.0%

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

      7. *-commutative 73.0%

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

      8. distribute-lft-neg-in 73.0%

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

      9. times-frac 73.0%

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

      10. metadata-eval 73.0%

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

      11. metadata-eval 73.0%

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

      12. associate--l+ 73.0%

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

      13. fma-define 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in y around inf 39.1%

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

      1. *-commutative 39.1%

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

   7. Simplified 39.1%

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

   if 3.45000000000000003e165 < x

   1. Initial program 63.5%

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

      1. remove-double-neg 63.5%

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

      2. distribute-lft-neg-out 63.5%

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

      3. distribute-frac-neg2 63.5%

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

      4. distribute-frac-neg 63.5%

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

      5. neg-mul-1 63.5%

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

      6. distribute-lft-neg-out 63.5%

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

      7. *-commutative 63.5%

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

      8. distribute-lft-neg-in 63.5%

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

      9. times-frac 63.5%

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

      10. metadata-eval 63.5%

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

      11. metadata-eval 63.5%

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

      12. associate--l+ 63.5%

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

      13. fma-define 63.5%

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

   3. Simplified 63.5%

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

      1. prod-diff 63.5%

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

      2. fma-neg 63.5%

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

      3. difference-of-squares 63.7%

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

      4. fma-define 74.1%

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

      5. pow2 74.1%

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

   6. Applied egg-rr 74.1%

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

   8. Applied egg-rr 45.4%

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

   9. Taylor expanded in z around inf 8.1%

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

       1. unpow2 8.1%

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

       2. associate-*r* 8.1%

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

       3. *-commutative 8.1%

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

       4. associate-*r* 8.1%

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

       5. add-sqr-sqrt 12.6%

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

       6. div-inv 12.6%

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

   11. Applied egg-rr 12.6%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.45 \cdot 10^{+165}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 1.7× speedup

Math

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

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 (* 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) {
	return y_s * (0.5 * (y_m + (x * (x / y_m))));
}

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 * (0.5d0 * (y_m + (x * (x / y_m))))
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 * (0.5 * (y_m + (x * (x / y_m))));
}

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 * (0.5 * (y_m + (x * (x / y_m))))

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(0.5 * Float64(y_m + Float64(x * Float64(x / y_m)))))
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 * (0.5 * (y_m + (x * (x / y_m))));
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[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

   1. remove-double-neg 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. distribute-frac-neg2 71.9%

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

   4. distribute-frac-neg 71.9%

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

   5. neg-mul-1 71.9%

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

   6. distribute-lft-neg-out 71.9%

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

   7. *-commutative 71.9%

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

   8. distribute-lft-neg-in 71.9%

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

   9. times-frac 71.9%

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

   10. metadata-eval 71.9%

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

   11. metadata-eval 71.9%

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

   12. associate--l+ 71.9%

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

   13. fma-define 72.3%

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

3. Simplified 72.3%

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

5. Taylor expanded in z around 0 47.4%

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

   1. associate-*r/ 47.4%

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

   2. rem-square-sqrt 47.4%

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

   3. unpow2 47.4%

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

   4. unpow2 47.4%

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

   5. hypot-undefine 47.4%

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

   6. unpow2 47.4%

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

   7. unpow2 47.4%

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

   8. hypot-undefine 47.4%

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

   9. unpow2 47.4%

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

   10. associate-*r/ 47.4%

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

   11. *-commutative 47.4%

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

   12. metadata-eval 47.4%

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

   13. times-frac 47.4%

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

   14. associate-/l* 47.3%

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

7. Simplified 47.3%

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

8. Taylor expanded in x around 0 62.6%

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

   1. distribute-lft-out 62.6%

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

10. Simplified 62.6%

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

    1. unpow2 62.6%

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

    2. associate-/l* 66.5%

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

12. Applied egg-rr 66.5%

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

13. Final simplification 66.5%

    \[\leadsto 0.5 \cdot \left(y + x \cdot \frac{x}{y}\right) \]
14. Add Preprocessing

Alternative 5: 34.8% 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 71.9%

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

   1. remove-double-neg 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. distribute-frac-neg2 71.9%

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

   4. distribute-frac-neg 71.9%

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

   5. neg-mul-1 71.9%

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

   6. distribute-lft-neg-out 71.9%

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

   7. *-commutative 71.9%

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

   8. distribute-lft-neg-in 71.9%

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

   9. times-frac 71.9%

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

   10. metadata-eval 71.9%

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

   11. metadata-eval 71.9%

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

   12. associate--l+ 71.9%

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

   13. fma-define 72.3%

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

3. Simplified 72.3%

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

5. Taylor expanded in y around inf 35.4%

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

   1. *-commutative 35.4%

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

7. Simplified 35.4%

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

8. Final simplification 35.4%

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

Alternative 6: 2.8% accurate, 15.0× speedup

Math

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

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 z_m))

C

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

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 * z_m
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 * z_m;
}

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 * z_m

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 * z_m)
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 * z_m;
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 * z$95$m), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

   1. remove-double-neg 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. distribute-frac-neg2 71.9%

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

   4. distribute-frac-neg 71.9%

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

   5. neg-mul-1 71.9%

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

   6. distribute-lft-neg-out 71.9%

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

   7. *-commutative 71.9%

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

   8. distribute-lft-neg-in 71.9%

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

   9. times-frac 71.9%

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

   10. metadata-eval 71.9%

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

   11. metadata-eval 71.9%

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

   12. associate--l+ 71.9%

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

   13. fma-define 72.3%

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

3. Simplified 72.3%

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

   1. prod-diff 60.1%

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

   2. fma-neg 60.1%

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

   3. difference-of-squares 60.6%

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

   4. fma-define 63.7%

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

   5. pow2 63.7%

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

6. Applied egg-rr 63.7%

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

   1. add-sqr-sqrt 47.3%

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

   2. associate-/l* 47.3%

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

8. Applied egg-rr 65.8%

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

9. Taylor expanded in y around inf 39.9%

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

10. Taylor expanded in y around 0 3.0%

    \[\leadsto \color{blue}{z} \]

11. Final simplification 3.0%

    \[\leadsto z \]
12. 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 2024071 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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