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

Percentage Accurate: 68.8% → 99.9%

Time: 9.9s

Alternatives: 8

Speedup: 1.3×

• 41.6% 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: 68.8% 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

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

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 74.5%

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

3. Taylor expanded in x around 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. Final simplification 99.9%

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

Alternative 2: 39.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z (/ z y)) -0.5))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -1e-61)
     t_0
     (if (<= t_1 2e+112)
       (* y 0.5)
       (if (<= t_1 INFINITY) (* x (* 0.5 (/ x y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -1e-61) {
		tmp = t_0;
	} else if (t_1 <= 2e+112) {
		tmp = y * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (0.5 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * (z / y)) * -0.5;
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -1e-61) {
		tmp = t_0;
	} else if (t_1 <= 2e+112) {
		tmp = y * 0.5;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (0.5 * (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * (z / y)) * -0.5
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -1e-61:
		tmp = t_0
	elif t_1 <= 2e+112:
		tmp = y * 0.5
	elif t_1 <= math.inf:
		tmp = x * (0.5 * (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * Float64(z / y)) * -0.5)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-61)
		tmp = t_0;
	elseif (t_1 <= 2e+112)
		tmp = Float64(y * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(0.5 * Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * (z / y)) * -0.5;
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -1e-61)
		tmp = t_0;
	elseif (t_1 <= 2e+112)
		tmp = y * 0.5;
	elseif (t_1 <= Inf)
		tmp = x * (0.5 * (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $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, -1e-61], t$95$0, If[LessEqual[t$95$1, 2e+112], N[(y * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x * N[(0.5 * N[(x / y), $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))) < -1e-61 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 67.9

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.7

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

   7. Applied rewrites 32.7%

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

   9. Applied rewrites 35.0%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e112

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if 1.9999999999999999e112 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.1

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 38.8

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

   10. Applied rewrites 38.8%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{fma}\left(-z, \frac{z}{y}, y\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1e-61)
     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 <= -1e-61) {
		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(Float64(-z), Float64(z / 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 <= -1e-61)
		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, -1e-61], 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))) < -1e-61 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{y}^{2} - {z}^{2}}{y}} \]
   4. 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. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 64.4%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 81.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 64.2%

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

Alternative 4: 36.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -1e-61)
     (/ (* (* z z) -0.5) y)
     (if (<= t_0 2e+112) (* y 0.5) (* x (* 0.5 (/ x 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 <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else if (t_0 <= 2e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (0.5 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-1d-61)) then
        tmp = ((z * z) * (-0.5d0)) / y
    else if (t_0 <= 2d+112) then
        tmp = y * 0.5d0
    else
        tmp = x * (0.5d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else if (t_0 <= 2e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (0.5 * (x / 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 <= -1e-61:
		tmp = ((z * z) * -0.5) / y
	elif t_0 <= 2e+112:
		tmp = y * 0.5
	else:
		tmp = x * (0.5 * (x / 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 <= -1e-61)
		tmp = Float64(Float64(Float64(z * z) * -0.5) / y);
	elseif (t_0 <= 2e+112)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x / 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 <= -1e-61)
		tmp = ((z * z) * -0.5) / y;
	elseif (t_0 <= 2e+112)
		tmp = y * 0.5;
	else
		tmp = x * (0.5 * (x / 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, -1e-61], N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e+112], N[(y * 0.5), $MachinePrecision], N[(x * N[(0.5 * N[(x / 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))) < -1e-61

   1. Initial program 83.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 31.7

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

   5. Applied rewrites 31.7%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e112

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if 1.9999999999999999e112 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.4

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.5

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

   7. Applied rewrites 41.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 35.9

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

   10. Applied rewrites 35.9%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -1e-61)
     (/ (* (* z z) -0.5) y)
     (if (<= t_0 2e+112) (* y 0.5) (* x (* x (/ 0.5 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else if (t_0 <= 2e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-1d-61)) then
        tmp = ((z * z) * (-0.5d0)) / y
    else if (t_0 <= 2d+112) then
        tmp = y * 0.5d0
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else if (t_0 <= 2e+112) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -1e-61:
		tmp = ((z * z) * -0.5) / y
	elif t_0 <= 2e+112:
		tmp = y * 0.5
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-61)
		tmp = Float64(Float64(Float64(z * z) * -0.5) / y);
	elseif (t_0 <= 2e+112)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -1e-61)
		tmp = ((z * z) * -0.5) / y;
	elseif (t_0 <= 2e+112)
		tmp = y * 0.5;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-61], N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 2e+112], N[(y * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / 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))) < -1e-61

   1. Initial program 83.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 31.7

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

   5. Applied rewrites 31.7%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.9999999999999999e112

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if 1.9999999999999999e112 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 64.4

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 71.6

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 41.5

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

   7. Applied rewrites 41.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 35.9

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

   10. Applied rewrites 35.9%

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

   12. Applied rewrites 35.9%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+112}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -1e-61)
   (* (* z (/ z y)) -0.5)
   (* 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)) <= -1e-61) {
		tmp = (z * (z / y)) * -0.5;
	} 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)) <= -1e-61)
		tmp = Float64(Float64(z * Float64(z / y)) * -0.5);
	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], -1e-61], N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] * -0.5), $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))) < -1e-61

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 83.0

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate--l+ N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 83.0

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

   4. Applied rewrites 83.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.7

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

   7. Applied rewrites 31.7%

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

   9. Applied rewrites 32.5%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.8%

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

   3. Taylor expanded in 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 62.1%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -1e-61)
   (/ (* (* z z) -0.5) y)
   (* y 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)) <= (-1d-61)) then
        tmp = ((z * z) * (-0.5d0)) / y
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -1e-61) {
		tmp = ((z * z) * -0.5) / y;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -1e-61:
		tmp = ((z * z) * -0.5) / y
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -1e-61)
		tmp = Float64(Float64(Float64(z * z) * -0.5) / y);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -1e-61)
		tmp = ((z * z) * -0.5) / y;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -1e-61], N[(N[(N[(z * z), $MachinePrecision] * -0.5), $MachinePrecision] / y), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1e-61

   1. Initial program 83.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 31.7

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

   5. Applied rewrites 31.7%

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

   if -1e-61 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 29.7

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

   5. Applied rewrites 29.7%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(z \cdot z\right) \cdot -0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% 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 74.5%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 30.2

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

5. Applied rewrites 30.2%

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

6. Final simplification 30.2%

   \[\leadsto y \cdot 0.5 \]
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 2024221 
(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)))