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

Percentage Accurate: 70.0% → 99.9%

Time: 6.8s

Alternatives: 10

Speedup: 1.3×

• 41.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.0% 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

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

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 70.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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: 38.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+147}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \frac{z}{y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-107)
     (* (* (/ -0.5 y) z) z)
     (if (<= t_0 2e+147)
       (* 0.5 y)
       (if (<= t_0 INFINITY) (* (/ (* x x) y) 0.5) (* (* -0.5 (/ z y)) z))))))

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 <= -5e-107) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= 2e+147) {
		tmp = 0.5 * y;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x * x) / y) * 0.5;
	} else {
		tmp = (-0.5 * (z / y)) * z;
	}
	return tmp;
}

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 <= -5e-107) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= 2e+147) {
		tmp = 0.5 * y;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = ((x * x) / y) * 0.5;
	} else {
		tmp = (-0.5 * (z / y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-107:
		tmp = ((-0.5 / y) * z) * z
	elif t_0 <= 2e+147:
		tmp = 0.5 * y
	elif t_0 <= math.inf:
		tmp = ((x * x) / y) * 0.5
	else:
		tmp = (-0.5 * (z / y)) * z
	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 <= -5e-107)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	elseif (t_0 <= 2e+147)
		tmp = Float64(0.5 * y);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x * x) / y) * 0.5);
	else
		tmp = Float64(Float64(-0.5 * Float64(z / y)) * z);
	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 <= -5e-107)
		tmp = ((-0.5 / y) * z) * z;
	elseif (t_0 <= 2e+147)
		tmp = 0.5 * y;
	elseif (t_0 <= Inf)
		tmp = ((x * x) / y) * 0.5;
	else
		tmp = (-0.5 * (z / y)) * z;
	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, -5e-107], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e+147], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999971e-107

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

   8. Applied rewrites 45.0%

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

   10. Applied rewrites 45.0%

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

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e147

   1. Initial program 89.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 60.9

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

   5. Applied rewrites 60.9%

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

   if 2e147 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 47.6

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

   5. Applied rewrites 47.6%

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

   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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 54.0

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

   8. Applied rewrites 54.0%

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

Alternative 3: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-107} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{z}{y}, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 -5e-107) (not (<= t_0 INFINITY)))
     (* (fma (- z) (/ z y) 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 <= -5e-107) || !(t_0 <= ((double) INFINITY))) {
		tmp = fma(-z, (z / y), 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 <= -5e-107) || !(t_0 <= Inf))
		tmp = Float64(fma(Float64(-z), Float64(z / y), 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, -5e-107], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-z) * N[(z / y), $MachinePrecision] + y), $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))) < -4.99999999999999971e-107 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in x around 0

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

   8. Applied rewrites 72.8%

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

   10. Applied rewrites 72.8%

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

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0

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

      1. 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. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      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 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-107} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{z}{y}, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{y}, z, 0\right), z, y\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{z}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-107)
     (* (fma (fma (/ -1.0 y) z 0.0) z y) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (fma (- z) (/ z y) 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 <= -5e-107) {
		tmp = fma(fma((-1.0 / y), z, 0.0), z, y) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = fma(-z, (z / y), 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 <= -5e-107)
		tmp = Float64(fma(fma(Float64(-1.0 / y), z, 0.0), z, y) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	else
		tmp = Float64(fma(Float64(-z), Float64(z / y), 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[LessEqual[t$95$0, -5e-107], N[(N[(N[(N[(-1.0 / y), $MachinePrecision] * z + 0.0), $MachinePrecision] * z + y), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-z) * N[(z / y), $MachinePrecision] + y), $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))) < -4.99999999999999971e-107

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in x around inf

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 71.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{y}, z, 0\right), z, y\right) \cdot 0.5 \]

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0

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

      1. 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. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      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 71.2

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

   5. Applied rewrites 71.2%

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

   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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in x around 0

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

   8. Applied rewrites 78.3%

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

   10. Applied rewrites 78.3%

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

Alternative 5: 35.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-107} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 -5e-107) (not (<= t_0 INFINITY)))
     (* (* (/ -0.5 y) z) z)
     (* 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 <= -5e-107) || !(t_0 <= ((double) INFINITY))) {
		tmp = ((-0.5 / y) * z) * z;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

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 <= -5e-107) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = ((-0.5 / y) * z) * z;
	} else {
		tmp = 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 <= -5e-107) or not (t_0 <= math.inf):
		tmp = ((-0.5 / y) * z) * z
	else:
		tmp = 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 <= -5e-107) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	else
		tmp = 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 <= -5e-107) || ~((t_0 <= Inf)))
		tmp = ((-0.5 / y) * z) * z;
	else
		tmp = 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[Or[LessEqual[t$95$0, -5e-107], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $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))) < -4.99999999999999971e-107 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

   8. Applied rewrites 47.1%

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

   10. Applied rewrites 47.1%

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

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-107} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \frac{z}{y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-107)
     (* (* (/ -0.5 y) z) z)
     (if (<= t_0 INFINITY) (* 0.5 y) (* (* -0.5 (/ z y)) z)))))

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 <= -5e-107) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * y;
	} else {
		tmp = (-0.5 * (z / y)) * z;
	}
	return tmp;
}

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 <= -5e-107) {
		tmp = ((-0.5 / y) * z) * z;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y;
	} else {
		tmp = (-0.5 * (z / y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-107:
		tmp = ((-0.5 / y) * z) * z
	elif t_0 <= math.inf:
		tmp = 0.5 * y
	else:
		tmp = (-0.5 * (z / y)) * z
	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 <= -5e-107)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(-0.5 * Float64(z / y)) * z);
	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 <= -5e-107)
		tmp = ((-0.5 / y) * z) * z;
	elseif (t_0 <= Inf)
		tmp = 0.5 * y;
	else
		tmp = (-0.5 * (z / y)) * z;
	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, -5e-107], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * y), $MachinePrecision], N[(N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision] * z), $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))) < -4.99999999999999971e-107

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

   8. Applied rewrites 45.0%

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

   10. Applied rewrites 45.0%

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

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 54.0

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

   8. Applied rewrites 54.0%

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

Alternative 7: 50.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -5e-107)
   (* (* (/ -0.5 y) z) z)
   (* (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)) <= -5e-107) {
		tmp = ((-0.5 / y) * z) * z;
	} 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)) <= -5e-107)
		tmp = Float64(Float64(Float64(-0.5 / y) * z) * z);
	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], -5e-107], N[(N[(N[(-0.5 / y), $MachinePrecision] * z), $MachinePrecision] * z), $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))) < -4.99999999999999971e-107

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. Taylor expanded in z around inf

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

   8. Applied rewrites 45.0%

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

   10. Applied rewrites 45.0%

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

   if -4.99999999999999971e-107 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.1%

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

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

      8. distribute-lft-in N/A

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

      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 65.4

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

   5. Applied rewrites 65.4%

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

Alternative 8: 34.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.1 \cdot 10^{+245}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.1e+245) (* 0.5 y) (* 0.5 (sqrt (* y y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.1e+245) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * sqrt((y * 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 <= 8.1d+245) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.1e+245) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 8.1e+245:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * math.sqrt((y * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.1e+245)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * sqrt(Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 8.1e+245)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.1e+245], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.10000000000000023e245

   1. Initial program 72.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 30.6

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

   5. Applied rewrites 30.6%

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

   if 8.10000000000000023e245 < x

   1. Initial program 36.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 9.1

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

   5. Applied rewrites 9.1%

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

   7. Applied rewrites 17.3%

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

Alternative 9: 34.5% 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 70.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 29.4

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

5. Applied rewrites 29.4%

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

Alternative 10: 1.8% 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 70.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 29.4

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

5. Applied rewrites 29.4%

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

7. Applied rewrites 12.6%

   \[\leadsto 0.5 \cdot \sqrt{y \cdot y} \]

8. Taylor expanded in y around -inf

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

10. Applied rewrites 2.0%

    \[\leadsto -0.5 \cdot \color{blue}{y} \]
11. 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 2024339 
(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)))