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

Percentage Accurate: 69.5% → 99.9%

Time: 9.8s

Alternatives: 7

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.5% 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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (* 0.5 (fma (+ z_m x) (/ (- x z_m) y) y)))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 99.9%

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

Alternative 2: 69.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z\_m, \frac{z\_m}{-y}, y\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z\_m, \frac{x - z\_m}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -1e-128)
     (* 0.5 (fma z_m (/ z_m (- y)) y))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (* 0.5 (fma z_m (/ (- x z_m) y) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-128) {
		tmp = 0.5 * fma(z_m, (z_m / -y), y);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = 0.5 * fma(z_m, ((x - z_m) / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-128)
		tmp = Float64(0.5 * fma(z_m, Float64(z_m / Float64(-y)), y));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = Float64(0.5 * fma(z_m, Float64(Float64(x - z_m) / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-128], N[(0.5 * N[(z$95$m * N[(z$95$m / (-y)), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(z$95$m * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $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))) < -1.00000000000000005e-128

   1. Initial program 77.3%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.9

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

   5. Simplified 67.9%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\mathsf{neg}\left(z\right)}{y}, y\right)} \cdot \frac{1}{2} \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}{\mathsf{neg}\left(y\right)}}, y\right) \cdot \frac{1}{2} \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{z}}{\mathsf{neg}\left(y\right)}, y\right) \cdot \frac{1}{2} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{z}{\mathsf{neg}\left(y\right)}}, y\right) \cdot \frac{1}{2} \]

      14. neg-lowering-neg.f64 69.6

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

   7. Applied egg-rr 69.6%

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

   if -1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.4%

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

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 74.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 84.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{x - z}{y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{fma}\left(z\_m, \frac{z\_m}{-y}, y\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (fma z_m (/ z_m (- y)) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -1e-128)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x (/ x y) y)) t_0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = 0.5 * fma(z_m, (z_m / -y), y);
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -1e-128) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(0.5 * fma(z_m, Float64(z_m / Float64(-y)), y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-128)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(0.5 * N[(z$95$m * N[(z$95$m / (-y)), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-128], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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))) < -1.00000000000000005e-128 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.8%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 59.8

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

   5. Simplified 59.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. frac-2neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\mathsf{neg}\left(z\right)}{y}, y\right)} \cdot \frac{1}{2} \]

      11. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}{\mathsf{neg}\left(y\right)}}, y\right) \cdot \frac{1}{2} \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{z}}{\mathsf{neg}\left(y\right)}, y\right) \cdot \frac{1}{2} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{z}{\mathsf{neg}\left(y\right)}}, y\right) \cdot \frac{1}{2} \]

      14. neg-lowering-neg.f64 70.9

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

   7. Applied egg-rr 70.9%

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

   if -1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.4%

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

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 74.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-128}:\\ \;\;\;\;z\_m \cdot \left(-0.5 \cdot \frac{z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -1e-128)
     (* z_m (* -0.5 (/ z_m y)))
     (if (<= t_0 4e+131) (* 0.5 y) (* x (/ x (* y 2.0)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-128) {
		tmp = z_m * (-0.5 * (z_m / y));
	} else if (t_0 <= 4e+131) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-1d-128)) then
        tmp = z_m * ((-0.5d0) * (z_m / y))
    else if (t_0 <= 4d+131) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x / (y * 2.0d0))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-128) {
		tmp = z_m * (-0.5 * (z_m / y));
	} else if (t_0 <= 4e+131) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x / (y * 2.0));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -1e-128:
		tmp = z_m * (-0.5 * (z_m / y))
	elif t_0 <= 4e+131:
		tmp = 0.5 * y
	else:
		tmp = x * (x / (y * 2.0))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-128)
		tmp = Float64(z_m * Float64(-0.5 * Float64(z_m / y)));
	elseif (t_0 <= 4e+131)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -1e-128)
		tmp = z_m * (-0.5 * (z_m / y));
	elseif (t_0 <= 4e+131)
		tmp = 0.5 * y;
	else
		tmp = x * (x / (y * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-128], N[(z$95$m * N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+131], N[(0.5 * y), $MachinePrecision], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.00000000000000005e-128

   1. Initial program 77.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 27.9

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

   8. Simplified 27.9%

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

   if -1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 3.9999999999999996e131

   1. Initial program 90.2%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 57.7

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

   5. Simplified 57.7%

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

   if 3.9999999999999996e131 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 52.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 38.7

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

   5. Simplified 38.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 41.6

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

   7. Applied egg-rr 41.6%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 4 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;z\_m \cdot \left(-0.5 \cdot \frac{z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -1e-128)
   (* z_m (* -0.5 (/ z_m y)))
   (* 0.5 (fma x (/ x y) y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-128) {
		tmp = z_m * (-0.5 * (z_m / y));
	} else {
		tmp = 0.5 * fma(x, (x / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -1e-128)
		tmp = Float64(z_m * Float64(-0.5 * Float64(z_m / y)));
	else
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -1e-128], N[(z$95$m * N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $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))) < -1.00000000000000005e-128

   1. Initial program 77.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 27.9

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

   8. Simplified 27.9%

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

   if -1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.7%

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

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Simplified 71.7%

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

Alternative 6: 33.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -1 \cdot 10^{-128}:\\ \;\;\;\;z\_m \cdot \left(-0.5 \cdot \frac{z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -1e-128)
   (* z_m (* -0.5 (/ z_m y)))
   (* 0.5 y)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-128) {
		tmp = z_m * (-0.5 * (z_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)) <= (-1d-128)) then
        tmp = z_m * ((-0.5d0) * (z_m / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-128) {
		tmp = z_m * (-0.5 * (z_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if ((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-128:
		tmp = z_m * (-0.5 * (z_m / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -1e-128)
		tmp = Float64(z_m * Float64(-0.5 * Float64(z_m / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-128)
		tmp = z_m * (-0.5 * (z_m / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -1e-128], N[(z$95$m * N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $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))) < -1.00000000000000005e-128

   1. Initial program 77.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 27.9

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

   8. Simplified 27.9%

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

   if -1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.7%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 34.7

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

   5. Simplified 34.7%

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

Alternative 7: 34.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 y))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * y
end function

Java

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

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * y

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * y;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 67.2%

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

3. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 38.0

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

5. Simplified 38.0%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024198 
(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)))