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

Percentage Accurate: 69.3% → 96.2%

Time: 7.4s

Alternatives: 9

Speedup: 0.6×

• 42.6% 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: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 0.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (/ (- x z) y_m) (* 0.5 (+ z x)))
      (if (<= t_0 INFINITY)
        (* (fma (/ 1.0 (/ y_m x)) x y_m) 0.5)
        (* (/ (- y_m z) y_m) (* (+ z 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 t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((x - z) / y_m) * (0.5 * (z + x));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((1.0 / (y_m / x)), x, y_m) * 0.5;
	} else {
		tmp = ((y_m - z) / y_m) * ((z + 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)
	t_0 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(x - z) / y_m) * Float64(0.5 * Float64(z + x)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(1.0 / Float64(y_m / x)), x, y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(z + y_m) * 0.5));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(1.0 / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(z + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

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

   1. Initial program 78.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 58.4%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{x}}, x, y\right) \cdot 0.5 \]

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}}}{y \cdot 2} \]

      4. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}}}{y \cdot 2} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(y \cdot 2\right) \cdot \frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y \cdot 2}}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y \cdot 2}}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 85.0

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

   7. Applied rewrites 85.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;\frac{x - z}{y} \cdot \left(0.5 \cdot \left(z + x\right)\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{y}{x}}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \left(\left(z + y\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{z}{y\_m} \cdot \left(-0.5 \cdot z\right)\\ t_1 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ z y_m) (* -0.5 z)))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 4e+149)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (* (/ x y_m) x) 0.5) t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x / y_m) * x) * 0.5;
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y_m) * x) * 0.5;
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z)
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 4e+149:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = ((x / y_m) * x) * 0.5
	else:
		tmp = t_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(z / y_m) * Float64(-0.5 * z))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x / y_m) * x) * 0.5);
	else
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = ((x / y_m) * x) * 0.5;
	else
		tmp = t_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[(z / y$95$m), $MachinePrecision] * N[(-0.5 * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 4e+149], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 33.9%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e149

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 4.0000000000000002e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.3

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

   5. Applied rewrites 30.3%

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

   7. Applied rewrites 33.1%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;\frac{z}{y} \cdot \left(-0.5 \cdot z\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-0.5 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{z}{y\_m} \cdot \left(-0.5 \cdot z\right)\\ t_1 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{0.5}{y\_m} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ z y_m) (* -0.5 z)))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 4e+149)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (/ 0.5 y_m) (* x x)) t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (0.5 / y_m) * (x * x);
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z)
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 4e+149:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (0.5 / y_m) * (x * x)
	else:
		tmp = t_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(z / y_m) * Float64(-0.5 * z))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(0.5 / y_m) * Float64(x * x));
	else
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (0.5 / y_m) * (x * x);
	else
		tmp = t_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[(z / y$95$m), $MachinePrecision] * N[(-0.5 * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 4e+149], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 33.9%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e149

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   if 4.0000000000000002e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.3

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

   5. Applied rewrites 30.3%

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

   7. Applied rewrites 30.3%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;\frac{z}{y} \cdot \left(-0.5 \cdot z\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot \left(-0.5 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 0.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (/ (- x z) y_m) (* 0.5 (+ z x)))
      (if (<= t_0 INFINITY)
        (* (fma (/ x y_m) x y_m) 0.5)
        (* (/ (- y_m z) y_m) (* (+ z 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 t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((x - z) / y_m) * (0.5 * (z + x));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	} else {
		tmp = ((y_m - z) / y_m) * ((z + 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)
	t_0 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(x - z) / y_m) * Float64(0.5 * Float64(z + x)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(z + y_m) * 0.5));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(z + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

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

   1. Initial program 78.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 58.4%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\color{blue}{\frac{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}}}{y \cdot 2} \]

      4. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}}}{y \cdot 2} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(y \cdot 2\right) \cdot \frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y \cdot 2}}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{y \cdot 2}}{\frac{\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right) + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) + \left(x \cdot x + y \cdot y\right) \cdot \left(z \cdot z\right)\right)}{{\left(x \cdot x + y \cdot y\right)}^{3} - {\left(z \cdot z\right)}^{3}}}} \]

   4. Applied rewrites 27.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 85.0

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

   7. Applied rewrites 85.0%

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

4. Final simplification 68.3%

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

Alternative 5: 96.4% accurate, 0.3× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ (- x z) y_m) (* 0.5 (+ z x))))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 INFINITY) (* (fma (/ x y_m) x y_m) 0.5) t_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 - z) / y_m) * (0.5 * (z + x));
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	} else {
		tmp = t_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(x - z) / y_m) * Float64(0.5 * Float64(z + x)))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(0.5 * N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

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

   1. Initial program 78.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 58.4%

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

4. Final simplification 66.6%

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

Alternative 6: 63.1% accurate, 0.4× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (/ z y_m) (* -0.5 z)))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (* y_s (if (<= t_1 0.0) t_0 (if (<= t_1 INFINITY) (* 0.5 y_m) t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y_m;
	} else {
		tmp = t_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 = (z / y_m) * (-0.5 * z)
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * y_m
	else:
		tmp = t_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(z / y_m) * Float64(-0.5 * z))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * y_m);
	else
		tmp = t_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 = (z / y_m) * (-0.5 * z);
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * y_m;
	else
		tmp = t_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[(z / y$95$m), $MachinePrecision] * N[(-0.5 * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * y$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 33.9%

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

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

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 31.8

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

   5. Applied rewrites 31.8%

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

4. Final simplification 33.0%

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

Alternative 7: 63.1% accurate, 0.4× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* (/ -0.5 y_m) z) z))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (* y_s (if (<= t_1 0.0) t_0 (if (<= t_1 INFINITY) (* 0.5 y_m) t_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 = ((-0.5 / y_m) * z) * z;
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = t_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 = ((-0.5 / y_m) * z) * z;
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y_m;
	} else {
		tmp = t_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 = ((-0.5 / y_m) * z) * z
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * y_m
	else:
		tmp = t_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(-0.5 / y_m) * z) * z)
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * y_m);
	else
		tmp = t_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 = ((-0.5 / y_m) * z) * z;
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * y_m;
	else
		tmp = t_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[(-0.5 / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * y$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.5%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 33.9%

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

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

   1. Initial program 78.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 31.8

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

   5. Applied rewrites 31.8%

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

4. Final simplification 33.0%

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

Alternative 8: 93.1% accurate, 0.6× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) 0.0)
    (* (/ z y_m) (* -0.5 z))
    (* (fma (/ x y_m) x 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 * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= 0.0) {
		tmp = (z / y_m) * (-0.5 * z);
	} else {
		tmp = fma((x / y_m), x, 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 (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= 0.0)
		tmp = Float64(Float64(z / y_m) * Float64(-0.5 * z));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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[N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(z / y$95$m), $MachinePrecision] * N[(-0.5 * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 27.9

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

   5. Applied rewrites 27.9%

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

   7. Applied rewrites 29.5%

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

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

   1. Initial program 63.5%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 56.1%

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

4. Final simplification 43.8%

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

Alternative 9: 34.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 30.0

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

5. Applied rewrites 30.0%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.1× 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 2024288 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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