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

Percentage Accurate: 68.5% → 97.1%

Time: 6.9s

Alternatives: 9

Speedup: 1.3×

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% 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: 97.1% accurate, 0.3× speedup

Math

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

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 y_m)) z)
      (if (<= t_0 INFINITY)
        (* (fma (/ x_m y_m) x_m y_m) 0.5)
        (* (fma (- x_m z) (/ z y_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 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 / y_m)) * z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x_m / y_m), x_m, y_m) * 0.5;
	} else {
		tmp = fma((x_m - z), (z / y_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)
	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 * Float64(z / y_m)) * z);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x_m / y_m), x_m, y_m) * 0.5);
	else
		tmp = Float64(fma(Float64(x_m - z), Float64(z / y_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_] := 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 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x$95$m - z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision] + y$95$m), $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 83.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 32.7%

       \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{-1}{2} \cdot \frac{z}{y}\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 32.7%

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

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

   1. Initial program 76.9%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z + x, -1 \cdot \frac{z}{y}, y\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 96.3%

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

Alternative 2: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \frac{z}{y\_m}\right) \cdot z\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x\_m}{y\_m} \cdot x\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* -0.5 (/ z y_m)) z))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 2e+151)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (* (/ x_m y_m) x_m) 0.5) t_0))))))

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 = (-0.5 * (z / y_m)) * z;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_m / y_m) * x_m) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

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 = (-0.5 * (z / y_m)) * z;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_m / y_m) * x_m) * 0.5;
	} else {
		tmp = t_0;
	}
	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 = (-0.5 * (z / y_m)) * z
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+151:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = ((x_m / y_m) * x_m) * 0.5
	else:
		tmp = t_0
	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(-0.5 * Float64(z / y_m)) * z)
	t_1 = 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_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_m / y_m) * x_m) * 0.5);
	else
		tmp = t_0;
	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 = (-0.5 * (z / y_m)) * z;
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+151)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = ((x_m / y_m) * x_m) * 0.5;
	else
		tmp = t_0;
	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[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = 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$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * 0.5), $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 68.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 38.0%

       \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{-1}{2} \cdot \frac{z}{y}\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 38.0%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.00000000000000003e151

   1. Initial program 98.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 64.9

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

   5. Applied rewrites 64.9%

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

   if 2.00000000000000003e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 40.2

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

   5. Applied rewrites 40.2%

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

   7. Applied rewrites 42.3%

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

Alternative 3: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \frac{z}{y\_m}\right) \cdot z\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (* -0.5 (/ z y_m)) z))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 2e+151)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* x_m x_m) (+ y_m y_m)) t_0))))))

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 = (-0.5 * (z / y_m)) * z;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

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 = (-0.5 * (z / y_m)) * z;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+151) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	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 = (-0.5 * (z / y_m)) * z
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+151:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = t_0
	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(-0.5 * Float64(z / y_m)) * z)
	t_1 = 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_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	else
		tmp = t_0;
	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 = (-0.5 * (z / y_m)) * z;
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+151)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x_m * x_m) / (y_m + y_m);
	else
		tmp = t_0;
	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[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = 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$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $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 68.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 38.0%

       \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{-1}{2} \cdot \frac{z}{y}\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 38.0%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.00000000000000003e151

   1. Initial program 98.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 64.9

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

   5. Applied rewrites 64.9%

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

   if 2.00000000000000003e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 71.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 31.3

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

   5. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 40.2

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

   10. Applied rewrites 40.2%

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

Alternative 4: 95.9% accurate, 0.3× speedup

Math

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

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 y_m)) z)
      (if (<= t_0 INFINITY)
        (* (fma (/ x_m y_m) x_m y_m) 0.5)
        (* (fma (- z) (/ z y_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 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 / y_m)) * z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x_m / y_m), x_m, y_m) * 0.5;
	} else {
		tmp = fma(-z, (z / y_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)
	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 * Float64(z / y_m)) * z);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x_m / y_m), x_m, y_m) * 0.5);
	else
		tmp = Float64(fma(Float64(-z), Float64(z / y_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_] := 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 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-z) * N[(z / y$95$m), $MachinePrecision] + y$95$m), $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 83.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 32.7%

       \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{-1}{2} \cdot \frac{z}{y}\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 32.7%

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

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

   1. Initial program 76.9%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(z + x, -1 \cdot \frac{z}{y}, y\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 96.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(-1 \cdot z, \frac{z}{y}, y\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 89.2%

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

Alternative 5: 93.0% accurate, 0.3× speedup

Math

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

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 (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
      (* (* -0.5 (/ z y_m)) z)
      (* (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 t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-0.5 * (z / y_m)) * z;
	} 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)
	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) || !(t_0 <= Inf))
		tmp = Float64(Float64(-0.5 * Float64(z / y_m)) * z);
	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_] := 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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(-0.5 * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * z), $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 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 68.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 38.0%

       \[\leadsto \left(\left(0 - \frac{0.5}{y}\right) \cdot z\right) \cdot \color{blue}{z} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{-1}{2} \cdot \frac{z}{y}\right) \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 38.0%

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

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

   1. Initial program 76.9%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

4. Final simplification 50.8%

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

Alternative 6: 51.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 8.6 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \end{array} \end{array} \]

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) 8.6e+176) (* 0.5 y_m) (/ (* x_m x_m) (+ y_m 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) <= 8.6e+176) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + 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) <= 8.6d+176) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x_m * x_m) / (y_m + 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) <= 8.6e+176) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + 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) <= 8.6e+176:
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * x_m) / (y_m + 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(x_m * x_m) <= 8.6e+176)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + 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) <= 8.6e+176)
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * x_m) / (y_m + 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[(x$95$m * x$95$m), $MachinePrecision], 8.6e+176], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.60000000000000051e176

   1. Initial program 71.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 47.3

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

   5. Applied rewrites 47.3%

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

   if 8.60000000000000051e176 < (*.f64 x x)

   1. Initial program 73.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 14.9

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

   5. Applied rewrites 14.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 14.9

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 7: 99.9% accurate, 1.2× speedup

Math

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

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 (* (+ (* (/ (- x_m z) y_m) (+ z 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) {
	return y_s * (((((x_m - z) / y_m) * (z + x_m)) + y_m) * 0.5);
}

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 * (((((x_m - z) / y_m) * (z + x_m)) + y_m) * 0.5d0)
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 * (((((x_m - z) / y_m) * (z + x_m)) + y_m) * 0.5);
}

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 * (((((x_m - z) / y_m) * (z + x_m)) + y_m) * 0.5)

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(Float64(Float64(Float64(Float64(x_m - z) / y_m) * Float64(z + x_m)) + y_m) * 0.5))
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 * (((((x_m - z) / y_m) * (z + x_m)) + y_m) * 0.5);
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[(N[(N[(N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(z + x$95$m), $MachinePrecision]), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 8: 99.9% accurate, 1.3× speedup

Math

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

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 (* (fma (+ z x_m) (/ (- x_m z) y_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) {
	return y_s * (fma((z + x_m), ((x_m - z) / y_m), y_m) * 0.5);
}

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(fma(Float64(z + x_m), Float64(Float64(x_m - z) / y_m), y_m) * 0.5))
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[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 9: 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 71.8%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 33.7

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

5. Applied rewrites 33.7%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2024320 
(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)))