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

Percentage Accurate: 69.1% → 95.8%

Time: 7.5s

Alternatives: 6

Speedup: 0.6×

• 42.4% 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: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y\_m - \frac{z}{y\_m} \cdot z\right)\\ t_1 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* 0.5 (- y_m (* (/ z y_m) z))))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 INFINITY) (* (fma (/ x y_m) x y_m) 0.5) t_0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = 0.5 * (y_m - ((z / y_m) * z));
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(0.5 * Float64(y_m - Float64(Float64(z / y_m) * z)))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

Wolfram

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_] := Block[{t$95$0 = N[(0.5 * N[(y$95$m - N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]), $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 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 78.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-frac N/A

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

      11. unpow2 N/A

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

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

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 95.0%

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

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

   1. Initial program 72.5%

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

   3. Taylor expanded in z around 0

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

      1. *-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 99.6%

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

4. Final simplification 97.5%

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

Alternative 2: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{y\_m} \cdot z\right)\\ t_1 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{0.5}{y\_m} \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* (/ z y_m) z)))
        (t_1 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 5e+152)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (* (/ 0.5 y_m) x) x) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = -0.5 * ((z / y_m) * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((0.5 / y_m) * x) * x;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Java

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) {
	double t_0 = -0.5 * ((z / y_m) * z);
	double t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+152) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((0.5 / y_m) * x) * x;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = -0.5 * ((z / y_m) * z)
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e+152:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = ((0.5 / y_m) * x) * x
	else:
		tmp = t_0
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(-0.5 * Float64(Float64(z / y_m) * z))
	t_1 = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(0.5 / y_m) * x) * x);
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = -0.5 * ((z / y_m) * z);
	t_1 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+152)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = ((0.5 / y_m) * x) * x;
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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_] := Block[{t$95$0 = N[(-0.5 * N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(2.0 * y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+152], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(0.5 / y$95$m), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]), $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 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 8.8

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

   5. Applied rewrites 8.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.5

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

   8. Applied rewrites 82.5%

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

   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. *-commutative N/A

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

      2. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if 5e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 59.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 61.6

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

   8. Applied rewrites 61.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{y} \cdot z\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq \infty:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{y} \cdot z\right)\\ \end{array} \]
5. 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 2024230 
(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)))