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

Percentage Accurate: 68.7% → 99.8%

Time: 9.7s

Alternatives: 9

Speedup: 1.3×

• 41.9% 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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (* 0.5 (fma (- x z) (* (/ 1.0 y) (+ x z)) y)))

C

double code(double x, double y, double z) {
	return 0.5 * fma((x - z), ((1.0 / y) * (x + z)), y);
}

Julia

function code(x, y, z)
	return Float64(0.5 * fma(Float64(x - z), Float64(Float64(1.0 / y) * Float64(x + z)), y))
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

   11. associate-+r+ N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. unpow2 N/A

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

   15. unpow2 N/A

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

   16. difference-of-squares N/A

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

   17. associate-/l* N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 100.0%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(x - z, \color{blue}{\frac{1}{y} \cdot \left(x + z\right)}, y\right) \]
8. Add Preprocessing

Alternative 2: 37.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* z (* z (/ -0.5 y)))
     (if (<= t_0 2e+153)
       (* y 0.5)
       (if (<= t_0 INFINITY) (* (/ 0.5 y) (* x x)) (* z (* -0.5 (/ z y))))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = z * (z * (-0.5 / y));
	} else if (t_0 <= 2e+153) {
		tmp = y * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (0.5 / y) * (x * x);
	} else {
		tmp = z * (-0.5 * (z / 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 <= 0.0) {
		tmp = z * (z * (-0.5 / y));
	} else if (t_0 <= 2e+153) {
		tmp = y * 0.5;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (0.5 / y) * (x * x);
	} else {
		tmp = z * (-0.5 * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = z * (z * (-0.5 / y))
	elif t_0 <= 2e+153:
		tmp = y * 0.5
	elif t_0 <= math.inf:
		tmp = (0.5 / y) * (x * x)
	else:
		tmp = z * (-0.5 * (z / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	elseif (t_0 <= 2e+153)
		tmp = Float64(y * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(0.5 / y) * Float64(x * x));
	else
		tmp = Float64(z * Float64(-0.5 * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = z * (z * (-0.5 / y));
	elseif (t_0 <= 2e+153)
		tmp = y * 0.5;
	elseif (t_0 <= Inf)
		tmp = (0.5 / y) * (x * x);
	else
		tmp = z * (-0.5 * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(y * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(0.5 / y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(-0.5 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 34.7%

       \[\leadsto z \cdot \left(\left(z \cdot -0.5\right) \cdot \color{blue}{\frac{1}{y}}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 34.7%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e153

   1. Initial program 97.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 71.9

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

   5. Applied rewrites 71.9%

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

   if 2e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 68.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 33.4

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

   5. Applied rewrites 33.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 33.4

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

   7. Applied rewrites 33.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 54.3

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

   8. Applied rewrites 54.3%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (fma z (/ z (- y)) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 2e-98)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x (* x (/ 1.0 y)) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * fma(z, (z / -y), y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 2e-98) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x * (1.0 / y)), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * fma(z, Float64(z / Float64(-y)), 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 <= 2e-98)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x * Float64(1.0 / y)), y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(z * N[(z / (-y)), $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, 2e-98], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(x * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.99999999999999988e-98 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 1.99999999999999988e-98 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.1%

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

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      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 66.2%

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

   7. Applied rewrites 66.2%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, \frac{1}{y} \cdot \color{blue}{x}, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (fma z (/ z (- y)) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 2e-98)
     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 = 0.5 * fma(z, (z / -y), y);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 2e-98) {
		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(0.5 * fma(z, Float64(z / Float64(-y)), 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 <= 2e-98)
		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[(0.5 * N[(z * N[(z / (-y)), $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, 2e-98], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.99999999999999988e-98 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

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

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. lower-/.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 70.1

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

   8. Applied rewrites 70.1%

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

   if 1.99999999999999988e-98 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.1%

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

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      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 66.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{z}{-y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 34.7%

       \[\leadsto z \cdot \left(\left(z \cdot -0.5\right) \cdot \color{blue}{\frac{1}{y}}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 34.7%

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

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

   1. Initial program 75.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 39.9

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

   5. Applied rewrites 39.9%

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

   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 54.3

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

   8. Applied rewrites 54.3%

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

4. Final simplification 38.9%

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

Alternative 6: 34.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-0.5 * (z / y));
	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.POSITIVE_INFINITY) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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 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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 38.0

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

   8. Applied rewrites 38.0%

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

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

   1. Initial program 75.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 39.9

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

   5. Applied rewrites 39.9%

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

4. Final simplification 38.9%

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

Alternative 7: 49.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) 0.0)
   (* 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)) <= 0.0) {
		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)) <= 0.0)
		tmp = Float64(z * Float64(z * Float64(-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], 0.0], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   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 z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 34.7%

       \[\leadsto z \cdot \left(\left(z \cdot -0.5\right) \cdot \color{blue}{\frac{1}{y}}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 34.7%

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

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

   1. Initial program 62.5%

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

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      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 62.9%

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

4. Final simplification 50.1%

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

Alternative 8: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y} + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

   11. associate-+r+ N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. unpow2 N/A

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

   15. unpow2 N/A

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

   16. difference-of-squares N/A

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

   17. associate-/l* N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 9: 34.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 36.0

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

5. Applied rewrites 36.0%

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

6. Final simplification 36.0%

   \[\leadsto y \cdot 0.5 \]
7. 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 2024219 
(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)))