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

Percentage Accurate: 69.1% → 93.9%

Time: 5.5s

Alternatives: 6

Speedup: 0.6×

• 41.8% 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: 69.1% 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: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(z + x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + 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 81.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

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

   1. Initial program 57.7%

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

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 69.1%

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

Alternative 2: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(-0.5 \cdot z\right) \cdot \frac{z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y\_m} \cdot \left(x\_m \cdot 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* -0.5 z) (/ z y_m))
      (if (<= t_0 5e+152) (* 0.5 y_m) (* (/ x_m y_m) (* x_m 0.5)))))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * z) * (z / y_m);
	} else if (t_0 <= 5e+152) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m / y_m) * (x_m * 0.5);
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= 0.0d0) then
        tmp = ((-0.5d0) * z) * (z / y_m)
    else if (t_0 <= 5d+152) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x_m / y_m) * (x_m * 0.5d0)
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (-0.5 * z) * (z / y_m);
	} else if (t_0 <= 5e+152) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m / y_m) * (x_m * 0.5);
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (-0.5 * z) * (z / y_m)
	elif t_0 <= 5e+152:
		tmp = 0.5 * y_m
	else:
		tmp = (x_m / y_m) * (x_m * 0.5)
	return y_s * tmp

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(-0.5 * z) * Float64(z / y_m));
	elseif (t_0 <= 5e+152)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m / y_m) * Float64(x_m * 0.5));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (-0.5 * z) * (z / y_m);
	elseif (t_0 <= 5e+152)
		tmp = 0.5 * y_m;
	else
		tmp = (x_m / y_m) * (x_m * 0.5);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+152], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m * 0.5), $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 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. 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 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 36.0%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5e152

   1. Initial program 99.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 64.6

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

   5. Applied rewrites 64.6%

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

   if 5e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 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 30.1

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

   5. Applied rewrites 30.1%

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

   7. Applied rewrites 34.7%

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

Alternative 3: 93.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + 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 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. 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 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 36.0%

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

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

   1. Initial program 57.7%

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

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 69.1%

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

Alternative 4: 61.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $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 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. 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 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 36.0%

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

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

   1. Initial program 57.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

Alternative 5: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{-99}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e-99)
    (* -0.5 (/ (* z z) y_m))
    (* 0.5 y_m))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-99) {
		tmp = -0.5 * ((z * z) / y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)) <= (-2d-99)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-99) {
		tmp = -0.5 * ((z * z) / y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z):
	tmp = 0
	if ((((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-99:
		tmp = -0.5 * ((z * z) / y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e-99)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-99)
		tmp = -0.5 * ((z * z) / y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e-99], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $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))) < -2e-99

   1. Initial program 83.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   if -2e-99 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

Alternative 6: 34.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $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$95$m_, y$95$m_, z_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 36.6

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

5. Applied rewrites 36.6%

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

Developer Target 1: 99.8% 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 2024317 
(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)))