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

Percentage Accurate: 69.4% → 99.9%

Time: 7.1s

Alternatives: 8

Speedup: 1.3×

• 41.3% 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: 69.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 68.2% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y y) (* x x)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_0 0.0)
     (* (- y (* (/ z_m y) z_m)) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (* (/ (+ z_m x) y) (- x z_m)) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (y - ((z_m / y) * z_m)) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = (((z_m + x) / y) * (x - z_m)) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(y - Float64(Float64(z_m / y) * z_m)) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(z_m + x) / y) * Float64(x - z_m)) * 0.5);
	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[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(y - N[(N[(z$95$m / y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(z$95$m + x), $MachinePrecision] / y), $MachinePrecision] * N[(x - z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 69.6%

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

   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 77.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.3%

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

   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. 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. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.3%

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

4. Final simplification 68.4%

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

Alternative 3: 68.5% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* (- y (* (/ z_m y) z_m)) 0.5))
        (t_1 (/ (- (+ (* y y) (* x x)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_1 0.0) t_0 (if (<= t_1 INFINITY) (* (fma (/ x y) x y) 0.5) t_0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (y - ((z_m / y) * z_m)) * 0.5;
	double t_1 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(y - Float64(Float64(z_m / y) * z_m)) * 0.5)
	t_1 = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	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[(N[(y - N[(N[(z$95$m / y), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $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))) < 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 63.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 69.9%

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

   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 77.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.3%

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

4. Final simplification 68.4%

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

Alternative 4: 37.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(y \cdot y + x \cdot x\right) - z\_m \cdot z\_m}{2 \cdot y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y y) (* x x)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_0 0.0)
     (* (* -0.5 (/ z_m y)) z_m)
     (if (<= t_0 5e+151) (* 0.5 y) (* (* (/ x y) x) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	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 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0d0 * y)
    if (t_0 <= 0.0d0) then
        tmp = ((-0.5d0) * (z_m / y)) * z_m
    else if (t_0 <= 5d+151) then
        tmp = 0.5d0 * y
    else
        tmp = ((x / y) * x) * 0.5d0
    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 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.5 * (z_m / y)) * z_m
	elif t_0 <= 5e+151:
		tmp = 0.5 * y
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m);
	elseif (t_0 <= 5e+151)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.5 * (z_m / y)) * z_m;
	elseif (t_0 <= 5e+151)
		tmp = 0.5 * y;
	else
		tmp = ((x / y) * x) * 0.5;
	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[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+151], N[(0.5 * y), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $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 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 30.9%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e151

   1. Initial program 99.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   if 5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 35.4%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot \frac{z}{y}\right) \cdot z\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(y \cdot y + x \cdot x\right) - z\_m \cdot z\_m}{2 \cdot y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y y) (* x x)) (* z_m z_m)) (* 2.0 y))))
   (if (<= t_0 0.0)
     (* (* -0.5 (/ z_m y)) z_m)
     (if (<= t_0 5e+151) (* 0.5 y) (* (* (/ 0.5 y) x) x)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	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 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0d0 * y)
    if (t_0 <= 0.0d0) then
        tmp = ((-0.5d0) * (z_m / y)) * z_m
    else if (t_0 <= 5d+151) then
        tmp = 0.5d0 * y
    else
        tmp = ((0.5d0 / y) * x) * x
    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 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.5 * (z_m / y)) * z_m
	elif t_0 <= 5e+151:
		tmp = 0.5 * y
	else:
		tmp = ((0.5 / y) * x) * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z_m * z_m)) / Float64(2.0 * y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m);
	elseif (t_0 <= 5e+151)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * x) * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.5 * (z_m / y)) * z_m;
	elseif (t_0 <= 5e+151)
		tmp = 0.5 * y;
	else
		tmp = ((0.5 / y) * x) * x;
	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[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+151], N[(0.5 * y), $MachinePrecision], N[(N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision] * x), $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 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 30.9%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e151

   1. Initial program 99.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   if 5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 35.4%

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{y} \cdot x, \color{blue}{x}, \frac{0}{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 35.4%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;\left(-0.5 \cdot \frac{z}{y}\right) \cdot z\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 0.6× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* y y) (* x x)) (* z_m z_m)) (* 2.0 y)) 0.0)
   (* (* -0.5 (/ z_m y)) z_m)
   (* (fma (/ x y) x y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((y * y) + (x * x)) - (z_m * z_m)) / (2.0 * y)) <= 0.0) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $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 77.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 30.9%

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

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

   1. Initial program 60.7%

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

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

      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 63.8%

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

4. Final simplification 47.6%

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

Alternative 7: 43.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 1.1e+91) (* (* (/ 0.5 y) x) x) (* 0.5 y)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 1.1e+91) {
		tmp = ((0.5 / y) * x) * x;
	} 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 (y <= 1.1d+91) then
        tmp = ((0.5d0 / y) * x) * x
    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 (y <= 1.1e+91) {
		tmp = ((0.5 / y) * x) * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if y <= 1.1e+91:
		tmp = ((0.5 / y) * x) * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 1.1e+91)
		tmp = Float64(Float64(Float64(0.5 / y) * x) * x);
	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 (y <= 1.1e+91)
		tmp = ((0.5 / y) * x) * x;
	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[y, 1.1e+91], N[(N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e91

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{y} \cdot x, \color{blue}{x}, \frac{0}{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 35.8%

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

   if 1.1e91 < y

   1. Initial program 31.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

Alternative 8: 34.3% 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 68.7%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 37.0

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

5. Applied rewrites 37.0%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024294 
(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)))