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

Percentage Accurate: 68.7% → 99.9%

Time: 10.2s

Alternatives: 7

Speedup: 1.3×

• 40.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% 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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (* 0.5 (fma (+ z_m x_m) (/ (- x_m z_m) y) y)))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. 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. Simplified 99.9%

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

Alternative 2: 38.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(z\_m \cdot -0.5\right) \cdot \frac{z\_m}{y}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x\_m \cdot \frac{0.5 \cdot x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* (* z_m -0.5) (/ z_m y)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -5e-109)
     t_0
     (if (<= t_1 2e+114)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* x_m (/ (* 0.5 x_m) y)) t_0)))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = (z_m * -0.5) * (z_m / y);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-109) {
		tmp = t_0;
	} else if (t_1 <= 2e+114) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_m * ((0.5 * x_m) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	double t_0 = (z_m * -0.5) * (z_m / y);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-109) {
		tmp = t_0;
	} else if (t_1 <= 2e+114) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_m * ((0.5 * x_m) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	t_0 = (z_m * -0.5) * (z_m / y)
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-109:
		tmp = t_0
	elif t_1 <= 2e+114:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = x_m * ((0.5 * x_m) / y)
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(z_m * -0.5) * Float64(z_m / y))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-109)
		tmp = t_0;
	elseif (t_1 <= 2e+114)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(x_m * Float64(Float64(0.5 * x_m) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
x_m = abs(x);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (z_m * -0.5) * (z_m / y);
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-109)
		tmp = t_0;
	elseif (t_1 <= 2e+114)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = x_m * ((0.5 * x_m) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m * -0.5), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-109], t$95$0, If[LessEqual[t$95$1, 2e+114], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-109 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.9%

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

   3. Taylor expanded in z around 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 36.1

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

   5. Simplified 36.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.7

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

   7. Applied egg-rr 39.7%

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

   if -5.0000000000000002e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e114

   1. Initial program 83.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 68.2

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

   5. Simplified 68.2%

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

   if 2e114 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 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 37.5

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

   5. Simplified 37.5%

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

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 41.8

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

   7. Applied egg-rr 41.8%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(z\_m \cdot -0.5\right) \cdot \frac{z\_m}{y}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x\_m, \frac{x\_m}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.5, z\_m \cdot \frac{z\_m}{y \cdot -2}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-109)
     (* (* z_m -0.5) (/ z_m y))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x_m (/ x_m y) y))
       (fma y 0.5 (* z_m (/ z_m (* y -2.0))))))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-109) {
		tmp = (z_m * -0.5) * (z_m / y);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x_m, (x_m / y), y);
	} else {
		tmp = fma(y, 0.5, (z_m * (z_m / (y * -2.0))));
	}
	return tmp;
}

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-109)
		tmp = Float64(Float64(z_m * -0.5) * Float64(z_m / y));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x_m, Float64(x_m / y), y));
	else
		tmp = fma(y, 0.5, Float64(z_m * Float64(z_m / Float64(y * -2.0))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-109], N[(N[(z$95$m * -0.5), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x$95$m * N[(x$95$m / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(y * 0.5 + N[(z$95$m * N[(z$95$m / N[(y * -2.0), $MachinePrecision]), $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))) < -5.0000000000000002e-109

   1. Initial program 81.0%

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

   3. Taylor expanded in z around inf

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

      1. 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 37.5

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

   5. Simplified 37.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.0

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

   7. Applied egg-rr 39.0%

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

   if -5.0000000000000002e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 67.6%

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

   3. Taylor expanded in z around 0

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

      1. *-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. Simplified 76.2%

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

   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. Simplified 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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 78.7

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

   8. Simplified 78.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. lift-neg.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 78.7

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-/.f64 N/A

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

      22. clear-num N/A

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

      23. un-div-inv N/A

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

      24. lower-/.f64 N/A

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

      25. div-inv N/A

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

      26. lower-*.f64 N/A

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

      27. metadata-eval 78.7

          \[\leadsto \mathsf{fma}\left(y, 0.5, z \cdot \frac{z}{y \cdot \color{blue}{-2}}\right) \]

   10. Applied egg-rr 78.7%

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

Alternative 4: 53.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(z\_m \cdot -0.5\right) \cdot \frac{z\_m}{y}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x\_m, \frac{x\_m}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z\_m, \frac{-z\_m}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-109)
     (* (* z_m -0.5) (/ z_m y))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x_m (/ x_m y) y))
       (* 0.5 (fma z_m (/ (- z_m) y) y))))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-109) {
		tmp = (z_m * -0.5) * (z_m / y);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x_m, (x_m / y), y);
	} else {
		tmp = 0.5 * fma(z_m, (-z_m / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-109)
		tmp = Float64(Float64(z_m * -0.5) * Float64(z_m / y));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x_m, Float64(x_m / y), y));
	else
		tmp = Float64(0.5 * fma(z_m, Float64(Float64(-z_m) / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-109], N[(N[(z$95$m * -0.5), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x$95$m * N[(x$95$m / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(z$95$m * N[((-z$95$m) / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-109

   1. Initial program 81.0%

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

   3. Taylor expanded in z around inf

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

      1. 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 37.5

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

   5. Simplified 37.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.0

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

   7. Applied egg-rr 39.0%

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

   if -5.0000000000000002e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 67.6%

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

   3. Taylor expanded in z around 0

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

      1. *-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. Simplified 76.2%

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

   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. Simplified 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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 78.7

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

   8. Simplified 78.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(z \cdot -0.5\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(z, \frac{-z}{y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(z\_m \cdot -0.5\right) \cdot \frac{z\_m}{y}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* (* z_m -0.5) (/ z_m y)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -5e-109) t_0 (if (<= t_1 INFINITY) (* 0.5 y) t_0))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = (z_m * -0.5) * (z_m / y);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-109) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	double t_0 = (z_m * -0.5) * (z_m / y);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-109) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	t_0 = (z_m * -0.5) * (z_m / y)
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-109:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 0.5 * y
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(z_m * -0.5) * Float64(z_m / y))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-109)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
x_m = abs(x);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (z_m * -0.5) * (z_m / y);
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-109)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 0.5 * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m * -0.5), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-109], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-109 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.9%

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

   3. Taylor expanded in z around 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 36.1

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

   5. Simplified 36.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.7

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

   7. Applied egg-rr 39.7%

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

   if -5.0000000000000002e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 67.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 41.0

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

   5. Simplified 41.0%

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

Alternative 6: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\left(z\_m \cdot -0.5\right) \cdot \frac{z\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x\_m, \frac{x\_m}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0)) -5e-109)
   (* (* z_m -0.5) (/ z_m y))
   (* 0.5 (fma x_m (/ x_m y) y))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -5e-109) {
		tmp = (z_m * -0.5) * (z_m / y);
	} else {
		tmp = 0.5 * fma(x_m, (x_m / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -5e-109)
		tmp = Float64(Float64(z_m * -0.5) * Float64(z_m / y));
	else
		tmp = Float64(0.5 * fma(x_m, Float64(x_m / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -5e-109], N[(N[(z$95$m * -0.5), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x$95$m * N[(x$95$m / 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))) < -5.0000000000000002e-109

   1. Initial program 81.0%

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

   3. Taylor expanded in z around inf

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

      1. 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 37.5

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

   5. Simplified 37.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.0

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

   7. Applied egg-rr 39.0%

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

   if -5.0000000000000002e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 51.0%

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

   3. Taylor expanded in z around 0

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

      1. *-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. Simplified 69.7%

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

Alternative 7: 35.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ 0.5 \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m) :precision binary64 (* 0.5 y))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Fortran

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

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	return 0.5 * y

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 36.8

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

5. Simplified 36.8%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024212 
(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)))