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

Percentage Accurate: 69.4% → 93.9%

Time: 8.2s

Alternatives: 9

Speedup: 0.6×

• 41.3% 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: 69.4% 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

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z}{\frac{y\_m}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x}{\frac{-y\_m}{x}}, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* 0.5 (- y_m (/ z (/ y_m z))))
    (* (fma -1.0 (/ x (/ (- y_m) x)) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else {
		tmp = fma(-1.0, (x / (-y_m / x)), y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(0.5 * Float64(y_m - Float64(z / Float64(y_m / z))));
	else
		tmp = Float64(fma(-1.0, Float64(x / Float64(Float64(-y_m) / x)), y_m) * 0.5);
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(0.5 * N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(x / N[((-y$95$m) / x), $MachinePrecision]), $MachinePrecision] + 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))) < -2e-109

   1. Initial program 78.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 65.8%

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

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

   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 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 68.6%

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

   7. Applied rewrites 68.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{x}}, x, y\right) \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 68.6%

      \[\leadsto \mathsf{fma}\left(-1, \frac{x}{\frac{-y}{x}}, y\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x}{\frac{-y}{x}}, y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\left(\frac{z}{y\_m} \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

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 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_0 -2e-109)
      (* (* (/ z y_m) z) -0.5)
      (if (<= t_0 5e+146) (* 0.5 y_m) (* (* (/ x y_m) x) 0.5))))))

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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= -2e-109) {
		tmp = ((z / y_m) * z) * -0.5;
	} else if (t_0 <= 5e+146) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0d0 * y_m)
    if (t_0 <= (-2d-109)) then
        tmp = ((z / y_m) * z) * (-0.5d0)
    else if (t_0 <= 5d+146) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((x / y_m) * x) * 0.5d0
    end if
    code = y_s * tmp
end function

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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= -2e-109) {
		tmp = ((z / y_m) * z) * -0.5;
	} else if (t_0 <= 5e+146) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_0 <= -2e-109:
		tmp = ((z / y_m) * z) * -0.5
	elif t_0 <= 5e+146:
		tmp = 0.5 * y_m
	else:
		tmp = ((x / y_m) * x) * 0.5
	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(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_0 <= -2e-109)
		tmp = Float64(Float64(Float64(z / y_m) * z) * -0.5);
	elseif (t_0 <= 5e+146)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(x / y_m) * x) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_0 <= -2e-109)
		tmp = ((z / y_m) * z) * -0.5;
	elseif (t_0 <= 5e+146)
		tmp = 0.5 * y_m;
	else
		tmp = ((x / y_m) * x) * 0.5;
	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[(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$0, -2e-109], N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+146], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   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. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.5

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

   8. Applied rewrites 34.5%

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

   10. Applied rewrites 35.2%

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

   if -2e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e146

   1. Initial program 96.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   if 4.9999999999999999e146 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.5%

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

   7. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{x}}, x, y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   9. 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. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.1

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

   10. Applied rewrites 42.1%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\left(\frac{z}{y} \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 5 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\frac{z \cdot z}{y\_m} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

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 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m))))
   (*
    y_s
    (if (<= t_0 -2e-109)
      (* (/ (* z z) y_m) -0.5)
      (if (<= t_0 5e+146) (* 0.5 y_m) (* (* (/ x y_m) x) 0.5))))))

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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= -2e-109) {
		tmp = ((z * z) / y_m) * -0.5;
	} else if (t_0 <= 5e+146) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0d0 * y_m)
    if (t_0 <= (-2d-109)) then
        tmp = ((z * z) / y_m) * (-0.5d0)
    else if (t_0 <= 5d+146) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((x / y_m) * x) * 0.5d0
    end if
    code = y_s * tmp
end function

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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	double tmp;
	if (t_0 <= -2e-109) {
		tmp = ((z * z) / y_m) * -0.5;
	} else if (t_0 <= 5e+146) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * x) * 0.5;
	}
	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 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)
	tmp = 0
	if t_0 <= -2e-109:
		tmp = ((z * z) / y_m) * -0.5
	elif t_0 <= 5e+146:
		tmp = 0.5 * y_m
	else:
		tmp = ((x / y_m) * x) * 0.5
	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(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m))
	tmp = 0.0
	if (t_0 <= -2e-109)
		tmp = Float64(Float64(Float64(z * z) / y_m) * -0.5);
	elseif (t_0 <= 5e+146)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(x / y_m) * x) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m);
	tmp = 0.0;
	if (t_0 <= -2e-109)
		tmp = ((z * z) / y_m) * -0.5;
	elseif (t_0 <= 5e+146)
		tmp = 0.5 * y_m;
	else
		tmp = ((x / y_m) * x) * 0.5;
	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[(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$0, -2e-109], N[(N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+146], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.5

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

   5. Applied rewrites 34.5%

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

   if -2e-109 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e146

   1. Initial program 96.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 63.0

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

   5. Applied rewrites 63.0%

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

   if 4.9999999999999999e146 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 66.5%

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

   7. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{x}}, x, y\right) \cdot 0.5 \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   9. 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. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.1

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

   10. Applied rewrites 42.1%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\frac{z \cdot z}{y} \cdot -0.5\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{2 \cdot y} \leq 5 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \left(y\_m - \frac{z}{\frac{y\_m}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* 0.5 (- y_m (/ z (/ y_m z))))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = 0.5 * (y_m - (z / (y_m / z)));
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(0.5 * Float64(y_m - Float64(z / Float64(y_m / z))));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(0.5 * N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 65.8%

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

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

   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 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 68.6%

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

4. Final simplification 67.2%

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

Alternative 5: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\left(z + x\right) \cdot \frac{\left(x - z\right) \cdot 0.5}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* (+ z x) (/ (* (- x z) 0.5) y_m))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = (z + x) * (((x - z) * 0.5) / y_m);
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(Float64(z + x) * Float64(Float64(Float64(x - z) * 0.5) / y_m));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(N[(z + x), $MachinePrecision] * N[(N[(N[(x - z), $MachinePrecision] * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

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

   3. Taylor expanded in y around 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. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-+.f64 69.9

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

   5. Applied rewrites 69.9%

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

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

   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 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 68.6%

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

4. Final simplification 69.2%

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

Alternative 6: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y\_m}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* -0.5 (/ z (/ y_m z)))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = -0.5 * (z / (y_m / z));
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(-0.5 * Float64(z / Float64(y_m / z)));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(-0.5 * N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   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. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.5

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

   8. Applied rewrites 34.5%

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

   10. Applied rewrites 35.2%

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

   12. Applied rewrites 35.2%

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

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

   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 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 68.6%

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

4. Final simplification 52.6%

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

Alternative 7: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\left(\frac{z}{y\_m} \cdot z\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* (* (/ z y_m) z) -0.5)
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = ((z / y_m) * z) * -0.5;
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(Float64(Float64(z / y_m) * z) * -0.5);
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   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. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.5

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

   8. Applied rewrites 34.5%

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

   10. Applied rewrites 35.2%

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

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

   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 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 68.6%

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

4. Final simplification 52.5%

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

Alternative 8: 59.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{2 \cdot y\_m} \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\frac{z \cdot z}{y\_m} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* 2.0 y_m)) -2e-109)
    (* (/ (* z z) y_m) -0.5)
    (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = ((z * z) / y_m) * -0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0d0 * y_m)) <= (-2d-109)) then
        tmp = ((z * z) / y_m) * (-0.5d0)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

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 tmp;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109) {
		tmp = ((z * z) / y_m) * -0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	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):
	tmp = 0
	if ((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109:
		tmp = ((z * z) / y_m) * -0.5
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(2.0 * y_m)) <= -2e-109)
		tmp = Float64(Float64(Float64(z * z) / y_m) * -0.5);
	else
		tmp = Float64(0.5 * y_m);
	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)
	tmp = 0.0;
	if (((((y_m * y_m) + (x * x)) - (z * z)) / (2.0 * y_m)) <= -2e-109)
		tmp = ((z * z) / y_m) * -0.5;
	else
		tmp = 0.5 * y_m;
	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_] := N[(y$95$s * If[LessEqual[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], -2e-109], N[(N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision] * -0.5), $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-109

   1. Initial program 78.1%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.5

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

   5. Applied rewrites 34.5%

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

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

   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. *-commutative N/A

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

      2. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

4. Final simplification 34.3%

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

Alternative 9: 33.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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) {
	return y_s * (0.5 * y_m);
}

Python

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

Julia

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

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (0.5 * y_m);
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_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 32.8

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

5. Applied rewrites 32.8%

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

6. Final simplification 32.8%

   \[\leadsto 0.5 \cdot y \]
7. 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 2024235 
(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)))