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

Percentage Accurate: 69.4% → 99.8%

Time: 6.5s

Alternatives: 10

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: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 39.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y} \cdot \left(z \cdot -0.5\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z y) (* z -0.5)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 5e+149)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* (* (/ x y) x) 0.5) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) * (z * -0.5);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+149) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / y) * (z * -0.5);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+149) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) * (z * -0.5)
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e+149:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = ((x / y) * x) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / y) * Float64(z * -0.5))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+149)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) * (z * -0.5);
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+149)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = ((x / y) * x) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+149], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 32.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e149

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 4.9999999999999999e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.4

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

   5. Applied rewrites 34.4%

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

   7. Applied rewrites 40.6%

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

Alternative 3: 68.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
     (* (* (+ z x) (/ (- x z) y)) 0.5)
     (* (fma (/ x y) x y) 0.5))))

C

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

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y)) * 0.5);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $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 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 57.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. difference-of-squares N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 62.8

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

   5. Applied rewrites 62.8%

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

   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 80.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

4. Final simplification 61.9%

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

Alternative 4: 52.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(y - N[(N[(z / y), $MachinePrecision] * z), $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 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 28.5%

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

   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 80.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

   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

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

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 71.4%

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

Alternative 5: 36.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* -0.5 (/ (* z z) y))
     (if (<= t_0 5e+149) (* 0.5 y) (* (* (/ x y) x) 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (t_0 <= 5d+149) then
        tmp = 0.5d0 * y
    else
        tmp = ((x / y) * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -0.5 * ((z * z) / y)
	elif t_0 <= 5e+149:
		tmp = 0.5 * y
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (t_0 <= 5e+149)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -0.5 * ((z * z) / y);
	elseif (t_0 <= 5e+149)
		tmp = 0.5 * y;
	else
		tmp = ((x / y) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+149], 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 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e149

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 4.9999999999999999e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   7. Applied rewrites 37.9%

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

Alternative 6: 36.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* -0.5 (/ (* z z) y))
     (if (<= t_0 5e+149) (* 0.5 y) (* (* (/ 0.5 y) x) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (t_0 <= 5d+149) then
        tmp = 0.5d0 * y
    else
        tmp = ((0.5d0 / y) * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -0.5 * ((z * z) / y)
	elif t_0 <= 5e+149:
		tmp = 0.5 * y
	else:
		tmp = ((0.5 / y) * x) * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (t_0 <= 5e+149)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -0.5 * ((z * z) / y);
	elseif (t_0 <= 5e+149)
		tmp = 0.5 * y;
	else
		tmp = ((0.5 / y) * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+149], 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 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e149

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 4.9999999999999999e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   7. Applied rewrites 33.2%

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

   9. Applied rewrites 37.9%

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

Alternative 7: 34.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* -0.5 (/ (* z z) y))
     (if (<= t_0 5e+149) (* 0.5 y) (* (* x x) (/ 0.5 y))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (t_0 <= 5d+149) then
        tmp = 0.5d0 * y
    else
        tmp = (x * x) * (0.5d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 5e+149) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -0.5 * ((z * z) / y)
	elif t_0 <= 5e+149:
		tmp = 0.5 * y
	else:
		tmp = (x * x) * (0.5 / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (t_0 <= 5e+149)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -0.5 * ((z * z) / y);
	elseif (t_0 <= 5e+149)
		tmp = 0.5 * y;
	else
		tmp = (x * x) * (0.5 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+149], N[(0.5 * y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 / 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))) < 0.0

   1. Initial program 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e149

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 59.8

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

   5. Applied rewrites 59.8%

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

   if 4.9999999999999999e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   7. Applied rewrites 33.2%

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

Alternative 8: 49.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(z / y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $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 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 28.5%

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

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

   1. Initial program 65.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

Alternative 9: 31.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) 0.0)
   (* -0.5 (/ (* z z) y))
   (* 0.5 y)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)) <= 0.0d0) then
        tmp = (-0.5d0) * ((z * z) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if ((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= 0.0:
		tmp = -0.5 * ((z * z) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $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))) < 0.0

   1. Initial program 71.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 25.2

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

   5. Applied rewrites 25.2%

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

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

   1. Initial program 65.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 28.7

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

   5. Applied rewrites 28.7%

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

Alternative 10: 33.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 67.8%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 34.7

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

5. Applied rewrites 34.7%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024326 
(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)))