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

Percentage Accurate: 69.5% → 93.7%

Time: 6.4s

Alternatives: 9

Speedup: 0.6×

• 42.1% 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: 69.5% 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: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq 0:\\ \;\;\;\;\frac{\left(x + z\_m\right) \cdot 0.5}{\frac{y\_m}{x - z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) 0.0)
    (/ (* (+ x z_m) 0.5) (/ y_m (- x z_m)))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = ((x + z_m) * 0.5) / (y_m / (x - z_m));
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= 0.0)
		tmp = Float64(Float64(Float64(x + z_m) * 0.5) / Float64(y_m / Float64(x - z_m)));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(x + z$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(y$95$m / N[(x - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 67.3%

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

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

   1. Initial program 65.7%

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

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

   5. Applied rewrites 67.1%

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

Alternative 2: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{z\_m}{y\_m} \cdot -0.5\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+152} \lor \neg \left(t\_0 \leq 4 \cdot 10^{+292}\right):\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* (/ z_m y_m) -0.5) z_m)
      (if (or (<= t_0 2e+152) (not (<= t_0 4e+292)))
        (* 0.5 y_m)
        (* (* x x) (/ 0.5 y_m)))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else if ((t_0 <= 2e+152) || !(t_0 <= 4e+292)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) * (0.5 / y_m);
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = ((z_m / y_m) * (-0.5d0)) * z_m
    else if ((t_0 <= 2d+152) .or. (.not. (t_0 <= 4d+292))) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x * x) * (0.5d0 / y_m)
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else if ((t_0 <= 2e+152) || !(t_0 <= 4e+292)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) * (0.5 / y_m);
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((z_m / y_m) * -0.5) * z_m
	elif (t_0 <= 2e+152) or not (t_0 <= 4e+292):
		tmp = 0.5 * y_m
	else:
		tmp = (x * x) * (0.5 / y_m)
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m / y_m) * -0.5) * z_m);
	elseif ((t_0 <= 2e+152) || !(t_0 <= 4e+292))
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((z_m / y_m) * -0.5) * z_m;
	elseif ((t_0 <= 2e+152) || ~((t_0 <= 4e+292)))
		tmp = 0.5 * y_m;
	else
		tmp = (x * x) * (0.5 / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(z$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e+152], N[Not[LessEqual[t$95$0, 4e+292]], $MachinePrecision]], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e152 or 4.0000000000000001e292 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if 2.0000000000000001e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000001e292

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 47.1%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;\left(\frac{z}{y} \cdot -0.5\right) \cdot z\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+152} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 4 \cdot 10^{+292}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{z\_m}{y\_m} \cdot -0.5\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* (/ z_m y_m) -0.5) z_m)
      (if (<= t_0 2e+152) (* 0.5 y_m) (* (* (/ x y_m) x) 0.5))))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else if (t_0 <= 2e+152) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = ((z_m / y_m) * (-0.5d0)) * z_m
    else if (t_0 <= 2d+152) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((x / y_m) * x) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else if (t_0 <= 2e+152) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = ((z_m / y_m) * -0.5) * z_m
	elif t_0 <= 2e+152:
		tmp = 0.5 * y_m
	else:
		tmp = ((x / y_m) * x) * 0.5
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m / y_m) * -0.5) * z_m);
	elseif (t_0 <= 2e+152)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(x / y_m) * x) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	t_0 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((z_m / y_m) * -0.5) * z_m;
	elseif (t_0 <= 2e+152)
		tmp = 0.5 * y_m;
	else
		tmp = ((x / y_m) * x) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(z$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, 2e+152], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e152

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 55.0

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

   5. Applied rewrites 55.0%

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

   if 2.0000000000000001e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

   7. Applied rewrites 40.5%

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

Alternative 4: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq 0:\\ \;\;\;\;\left(\left(z\_m + x\right) \cdot 0.5\right) \cdot \frac{x - z\_m}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) 0.0)
    (* (* (+ z_m x) 0.5) (/ (- x z_m) y_m))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = ((z_m + x) * 0.5) * ((x - z_m) / y_m);
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= 0.0)
		tmp = Float64(Float64(Float64(z_m + x) * 0.5) * Float64(Float64(x - z_m) / y_m));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(z$95$m + x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

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

   1. Initial program 65.7%

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

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

   5. Applied rewrites 67.1%

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

Alternative 5: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -1 \cdot 10^{-44}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m}{\frac{y\_m}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -1e-44)
    (* -0.5 (/ z_m (/ y_m z_m)))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -1e-44) {
		tmp = -0.5 * (z_m / (y_m / z_m));
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -1e-44)
		tmp = Float64(-0.5 * Float64(z_m / Float64(y_m / z_m)));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -1e-44], N[(-0.5 * N[(z$95$m / N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $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))) < -9.99999999999999953e-45

   1. Initial program 73.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.7

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 31.4%

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

   if -9.99999999999999953e-45 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

   5. Applied rewrites 68.3%

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

Alternative 6: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -1 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{z\_m}{y\_m} \cdot -0.5\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -1e-44)
    (* (* (/ z_m y_m) -0.5) z_m)
    (* (fma (/ x y_m) x y_m) 0.5))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -1e-44) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -1e-44)
		tmp = Float64(Float64(Float64(z_m / y_m) * -0.5) * z_m);
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -1e-44], N[(N[(N[(z$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $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))) < -9.99999999999999953e-45

   1. Initial program 73.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.3%

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

   if -9.99999999999999953e-45 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 63.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. distribute-lft1-in N/A

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

      7. +-commutative N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

   5. Applied rewrites 68.3%

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

Alternative 7: 60.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq 0:\\ \;\;\;\;\left(\frac{z\_m}{y\_m} \cdot -0.5\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) 0.0)
    (* (* (/ z_m y_m) -0.5) z_m)
    (* 0.5 y_m))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)) <= 0.0d0) then
        tmp = ((z_m / y_m) * (-0.5d0)) * z_m
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = ((z_m / y_m) * -0.5) * z_m;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0:
		tmp = ((z_m / y_m) * -0.5) * z_m
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= 0.0)
		tmp = Float64(Float64(Float64(z_m / y_m) * -0.5) * z_m);
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0)
		tmp = ((z_m / y_m) * -0.5) * z_m;
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(z$95$m / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision] * z$95$m), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

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

   1. Initial program 65.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

Alternative 8: 60.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq 0:\\ \;\;\;\;\left(z\_m \cdot \frac{-0.5}{y\_m}\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) 0.0)
    (* (* z_m (/ -0.5 y_m)) z_m)
    (* 0.5 y_m))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = (z_m * (-0.5 / y_m)) * z_m;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0d0)) <= 0.0d0) then
        tmp = (z_m * ((-0.5d0) / y_m)) * z_m
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0) {
		tmp = (z_m * (-0.5 / y_m)) * z_m;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0:
		tmp = (z_m * (-0.5 / y_m)) * z_m
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= 0.0)
		tmp = Float64(Float64(z_m * Float64(-0.5 / y_m)) * z_m);
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z_m)
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= 0.0)
		tmp = (z_m * (-0.5 / y_m)) * z_m;
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(z$95$m * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 30.4%

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

   10. Applied rewrites 30.4%

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

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

   1. Initial program 65.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

Alternative 9: 34.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot y\_m\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m) :precision binary64 (* y_s (* 0.5 y_m)))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	return y_s * (0.5 * y_m);
}

Fortran

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = y_s * (0.5d0 * y_m)
end function

Java

z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z_m) {
	return y_s * (0.5 * y_m);
}

Python

z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z_m):
	return y_s * (0.5 * y_m)

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	return Float64(y_s * Float64(0.5 * y_m))
end

MATLAB

z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z_m)
	tmp = y_s * (0.5 * y_m);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 34.6

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

5. Applied rewrites 34.6%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024318 
(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)))