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

Percentage Accurate: 68.6% → 99.9%

Time: 10.1s

Alternatives: 7

Speedup: 1.3×

• 41.5% 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: 68.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 99.9%

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

Alternative 2: 40.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{z \cdot -0.5}{y}\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (* z -0.5) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-66)
     t_0
     (if (<= t_1 4e+151)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* x (/ x (* y 2.0))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= 4e+151) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x / (y * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= 4e+151) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x / (y * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * -0.5) / y)
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-66:
		tmp = t_0
	elif t_1 <= 4e+151:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = x * (x / (y * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * -0.5) / y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= 4e+151)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * -0.5) / y);
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= 4e+151)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = x * (x / (y * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $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, -5e-66], t$95$0, If[LessEqual[t$95$1, 4e+151], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $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))) < -4.99999999999999962e-66 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 33.7

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

   7. Applied rewrites 33.7%

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

   if -4.99999999999999962e-66 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.00000000000000007e151

   1. Initial program 84.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 4.00000000000000007e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 36.6

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

   7. Applied rewrites 36.6%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 4 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{z \cdot -0.5}{y}\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (* z -0.5) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-66)
     t_0
     (if (<= t_1 4e+151)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* (* x x) (/ 0.5 y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= 4e+151) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= 4e+151) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * -0.5) / y)
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-66:
		tmp = t_0
	elif t_1 <= 4e+151:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = (x * x) * (0.5 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * -0.5) / y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= 4e+151)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * -0.5) / y);
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= 4e+151)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = (x * x) * (0.5 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $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, -5e-66], t$95$0, If[LessEqual[t$95$1, 4e+151], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $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))) < -4.99999999999999962e-66 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 33.7

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

   7. Applied rewrites 33.7%

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

   if -4.99999999999999962e-66 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.00000000000000007e151

   1. Initial program 84.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 4.00000000000000007e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

      22. lower-/.f64 N/A

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

Alternative 4: 68.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 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 61.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 66.7

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

   8. Applied rewrites 66.7%

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

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

   1. Initial program 75.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

4. Final simplification 67.6%

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

Alternative 5: 36.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{z \cdot -0.5}{y}\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (* z -0.5) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-66) t_0 (if (<= t_1 INFINITY) (* 0.5 y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * -0.5) / y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-66) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * -0.5) / y)
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-66:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * -0.5) / y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * -0.5) / y);
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-66)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $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, -5e-66], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999962e-66 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 33.7

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

   7. Applied rewrites 33.7%

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

   if -4.99999999999999962e-66 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 72.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

Alternative 6: 51.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-66}:\\ \;\;\;\;z \cdot \frac{z \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -5e-66)
   (* z (/ (* z -0.5) y))
   (* 0.5 (fma x (/ x y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -5e-66) {
		tmp = z * ((z * -0.5) / y);
	} else {
		tmp = 0.5 * fma(x, (x / y), 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)) <= -5e-66)
		tmp = Float64(z * Float64(Float64(z * -0.5) / y));
	else
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	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-66], N[(z * N[(N[(z * -0.5), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999962e-66

   1. Initial program 78.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

   if -4.99999999999999962e-66 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 65.3%

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

Alternative 7: 34.3% 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 66.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 36.4

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

5. Applied rewrites 36.4%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024216 
(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)))