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

Percentage Accurate: 68.8% → 99.9%

Time: 9.8s

Alternatives: 10

Speedup: 1.3×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% 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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (* 0.5 (fma (+ x z_m) (/ (- x z_m) y) y)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * fma((x + z_m), ((x - z_m) / y), y);
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * fma(Float64(x + z_m), Float64(Float64(x - z_m) / y), y))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * N[(N[(x + z$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in x around 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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. div-sub N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

   11. associate-+r+ N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. unpow2 N/A

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

   15. unpow2 N/A

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

   16. difference-of-squares N/A

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

   17. associate-/l* N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 38.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := z\_m \cdot \left(z\_m \cdot \frac{-0.5}{y}\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+150}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (* z_m (/ -0.5 y))))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -5e-119)
     t_0
     (if (<= t_1 5e+150)
       (* y 0.5)
       (if (<= t_1 5e+299)
         (* x (* x (/ 0.5 y)))
         (if (<= t_1 INFINITY) (* y 0.5) t_0))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = z_m * (z_m * (-0.5 / y));
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-119) {
		tmp = t_0;
	} else if (t_1 <= 5e+150) {
		tmp = y * 0.5;
	} else if (t_1 <= 5e+299) {
		tmp = x * (x * (0.5 / y));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = z_m * (z_m * (-0.5 / y));
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-119) {
		tmp = t_0;
	} else if (t_1 <= 5e+150) {
		tmp = y * 0.5;
	} else if (t_1 <= 5e+299) {
		tmp = x * (x * (0.5 / y));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = z_m * (z_m * (-0.5 / y))
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-119:
		tmp = t_0
	elif t_1 <= 5e+150:
		tmp = y * 0.5
	elif t_1 <= 5e+299:
		tmp = x * (x * (0.5 / y))
	elif t_1 <= math.inf:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(z_m * Float64(z_m * Float64(-0.5 / y)))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-119)
		tmp = t_0;
	elseif (t_1 <= 5e+150)
		tmp = Float64(y * 0.5);
	elseif (t_1 <= 5e+299)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (t_1 <= Inf)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = z_m * (z_m * (-0.5 / y));
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-119)
		tmp = t_0;
	elseif (t_1 <= 5e+150)
		tmp = y * 0.5;
	elseif (t_1 <= 5e+299)
		tmp = x * (x * (0.5 / y));
	elseif (t_1 <= Inf)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(z$95$m * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-119], t$95$0, If[LessEqual[t$95$1, 5e+150], N[(y * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 5e+299], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * 0.5), $MachinePrecision], t$95$0]]]]]]

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))) < -4.99999999999999993e-119 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 54.2%

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

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 38.7%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.00000000000000009e150 or 5.0000000000000003e299 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   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 50.7

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

   5. Applied rewrites 50.7%

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

   if 5.00000000000000009e150 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000003e299

   1. Initial program 99.7%

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x - z, \color{blue}{\frac{1}{y} \cdot \left(x + z\right)}, y\right) \]

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 42.6

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

   10. Applied rewrites 42.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\left(x + z\_m\right) \cdot \left(0.5 \cdot \frac{x - z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z\_m \cdot \frac{z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-119)
     (* (+ x z_m) (* 0.5 (/ (- x z_m) y)))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (* 0.5 (- y (* z_m (/ z_m y))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-119) {
		tmp = (x + z_m) * (0.5 * ((x - z_m) / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = 0.5 * (y - (z_m * (z_m / y)));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-119)
		tmp = Float64(Float64(x + z_m) * Float64(0.5 * Float64(Float64(x - z_m) / y)));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z_m * Float64(z_m / y))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-119], N[(N[(x + z$95$m), $MachinePrecision] * N[(0.5 * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999993e-119

   1. Initial program 75.5%

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x - z, \color{blue}{\frac{1}{y} \cdot \left(x + z\right)}, y\right) \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-/l* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. div-sub N/A

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

      18. lower-/.f64 N/A

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

      19. lower--.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

   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

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

   7. Applied rewrites 84.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\left(x + z\right) \cdot \left(0.5 \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\_m\right) \cdot \frac{x + z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z\_m \cdot \frac{z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-119)
     (* 0.5 (* (- x z_m) (/ (+ x z_m) y)))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (* 0.5 (- y (* z_m (/ z_m y))))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-119) {
		tmp = 0.5 * ((x - z_m) * ((x + z_m) / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = 0.5 * (y - (z_m * (z_m / y)));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-119)
		tmp = Float64(0.5 * Float64(Float64(x - z_m) * Float64(Float64(x + z_m) / y)));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z_m * Float64(z_m / y))));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-119], N[(0.5 * N[(N[(x - z$95$m), $MachinePrecision] * N[(N[(x + z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999993e-119

   1. Initial program 75.5%

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.9%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

   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

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

   7. Applied rewrites 84.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \left(y - z\_m \cdot \frac{z\_m}{y}\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (- y (* z_m (/ z_m y)))))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -5e-119)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x (/ x y) y)) t_0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = 0.5 * (y - (z_m * (z_m / y)));
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-119) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(0.5 * Float64(y - Float64(z_m * Float64(z_m / y))))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-119)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(0.5 * N[(y - N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-119], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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))) < -4.99999999999999993e-119 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 54.2%

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 73.0%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \left(-\frac{z\_m}{y}\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\_m\right) \cdot \frac{z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-119)
     (* 0.5 (* z_m (- (/ z_m y))))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (* 0.5 (* (- x z_m) (/ z_m y)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-119) {
		tmp = 0.5 * (z_m * -(z_m / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = 0.5 * ((x - z_m) * (z_m / y));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-119)
		tmp = Float64(0.5 * Float64(z_m * Float64(-Float64(z_m / y))));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = Float64(0.5 * Float64(Float64(x - z_m) * Float64(z_m / y)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-119], N[(0.5 * N[(z$95$m * (-N[(z$95$m / y), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x - z$95$m), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999993e-119

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.3%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 60.1%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(-\frac{z}{y}\right)\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \left(-\frac{z\_m}{y}\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \left(z\_m \cdot \frac{-0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -5e-119)
     (* 0.5 (* z_m (- (/ z_m y))))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (* z_m (* z_m (/ -0.5 y)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-119) {
		tmp = 0.5 * (z_m * -(z_m / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = z_m * (z_m * (-0.5 / y));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-119)
		tmp = Float64(0.5 * Float64(z_m * Float64(-Float64(z_m / y))));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = Float64(z_m * Float64(z_m * Float64(-0.5 / y)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-119], N[(0.5 * N[(z$95$m * (-N[(z$95$m / y), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(z$95$m * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999993e-119

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 32.3%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

   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 z around inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 35.3

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

   5. Applied rewrites 35.3%

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

   7. Applied rewrites 55.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(-\frac{z}{y}\right)\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := z\_m \cdot \left(z\_m \cdot \frac{-0.5}{y}\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (* z_m (/ -0.5 y))))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -5e-119)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x (/ x y) y)) t_0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = z_m * (z_m * (-0.5 / y));
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-119) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(z_m * Float64(z_m * Float64(-0.5 / y)))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-119)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(z$95$m * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-119], t$95$0, If[LessEqual[t$95$1, Infinity], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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))) < -4.99999999999999993e-119 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 54.2%

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

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 38.7%

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

   if -4.99999999999999993e-119 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 74.7%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

      11. *-inverses N/A

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

      12. *-rgt-identity N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. associate-/r/ N/A

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

      17. unpow2 N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. *-rgt-identity N/A

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

   5. Applied rewrites 68.9%

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

Alternative 9: 43.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 6.8e+25) (* x (* x (/ 0.5 y))) (* y 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 6.8e+25) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 6.8d+25) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 6.8e+25) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if y <= 6.8e+25:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = y * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 6.8e+25)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (y <= 6.8e+25)
		tmp = x * (x * (0.5 / y));
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[y, 6.8e+25], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999967e25

   1. Initial program 70.8%

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. associate-+r+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. unpow2 N/A

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

      15. unpow2 N/A

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

      16. difference-of-squares N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(x - z, \color{blue}{\frac{1}{y} \cdot \left(x + z\right)}, y\right) \]

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 33.1

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

   10. Applied rewrites 33.1%

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

   if 6.79999999999999967e25 < y

   1. Initial program 45.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y \cdot 0.5 \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* y 0.5))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return y * 0.5;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = y * 0.5d0
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return y * 0.5;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return y * 0.5

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(y * 0.5)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = y * 0.5;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 40.2

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

5. Applied rewrites 40.2%

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

6. Final simplification 40.2%

   \[\leadsto y \cdot 0.5 \]
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 2024232 
(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)))