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

Percentage Accurate: 69.2% → 99.9%

Time: 13.8s

Alternatives: 7

Speedup: 1.3×

• 41.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, \mathsf{fma}\left(z + x, \frac{x - z}{y}, y\right), 0\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma 0.5 (fma (+ z x) (/ (- x z) y) y) 0.0))

C

double code(double x, double y, double z) {
	return fma(0.5, fma((z + x), ((x - z) / y), y), 0.0);
}

Julia

function code(x, y, z)
	return fma(0.5, fma(Float64(z + x), Float64(Float64(x - z) / y), y), 0.0)
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision] + 0.0), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(z + x, \frac{x - z}{y}, y\right), 0\right)} \]
6. Add Preprocessing

Alternative 2: 39.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+148}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ -0.5 y))))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-136)
     t_0
     (if (<= t_1 4e+148)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* x (/ x (* y 2.0))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-136) {
		tmp = t_0;
	} else if (t_1 <= 4e+148) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x / (y * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (z * (-0.5 / y));
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-136) {
		tmp = t_0;
	} else if (t_1 <= 4e+148) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x / (y * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * (-0.5 / y))
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= -5e-136:
		tmp = t_0
	elif t_1 <= 4e+148:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = x * (x / (y * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * Float64(-0.5 / y)))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-136)
		tmp = t_0;
	elseif (t_1 <= 4e+148)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * (-0.5 / y));
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -5e-136)
		tmp = t_0;
	elseif (t_1 <= 4e+148)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = x * (x / (y * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * 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 * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-136], t$95$0, If[LessEqual[t$95$1, 4e+148], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $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))) < -5.0000000000000002e-136 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.2%

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

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

      11. +-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left({z}^{2} + 0\right)} \cdot \frac{-1}{2}}{y} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot z} + 0\right) \cdot \frac{-1}{2}}{y} \]

      13. accelerator-lowering-fma.f64 34.2

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

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, z, 0\right) \cdot -0.5}{y}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 36.4

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

   7. Applied egg-rr 36.4%

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

   if -5.0000000000000002e-136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e148

   1. Initial program 97.1%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 55.6

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

   5. Simplified 55.6%

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

   if 4.0000000000000002e148 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 67.4%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} + 0}}{y \cdot 2} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} + 0}{y \cdot 2} \]

      3. accelerator-lowering-fma.f64 33.0

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}{y \cdot 2} \]

   5. Simplified 33.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}{y \cdot 2} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 36.4

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

   7. Applied egg-rr 36.4%

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

Alternative 3: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;\left(z + x\right) \cdot \left(\left(x - z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y - z \cdot \frac{z}{y}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-136)
     (* (+ z x) (* (- x z) (/ 0.5 y)))
     (if (<= t_0 INFINITY)
       (* 0.5 (fma x (/ x y) y))
       (fma 0.5 (- y (* z (/ z y))) 0.0)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-136) {
		tmp = (z + x) * ((x - z) * (0.5 / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = fma(0.5, (y - (z * (z / y))), 0.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-136)
		tmp = Float64(Float64(z + x) * Float64(Float64(x - z) * Float64(0.5 / y)));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	else
		tmp = fma(0.5, Float64(y - Float64(z * Float64(z / y))), 0.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-136], N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] * N[(0.5 / 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 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-136

   1. Initial program 76.2%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} + 0\right)} - z \cdot z}{y \cdot 2} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + 0\right) - z \cdot z}{y \cdot 2} \]

      3. accelerator-lowering-fma.f64 60.1

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)} - z \cdot z}{y \cdot 2} \]

   5. Simplified 60.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)} - z \cdot z}{y \cdot 2} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot 2}{\left(x \cdot x + 0\right) - z \cdot z}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot 2}{\left(x \cdot x + 0\right) - z \cdot z}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot 2}{\left(x \cdot x + 0\right) - z \cdot z}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot 2}}{\left(x \cdot x + 0\right) - z \cdot z}} \]

      5. +-rgt-identity N/A

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

      6. difference-of-squares N/A

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

      7. +-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. --lowering--.f64 60.1

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

   7. Applied egg-rr 60.1%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

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

      10. --lowering--.f64 N/A

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

      11. +-lowering-+.f64 65.0

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

   9. Applied egg-rr 65.0%

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

   if -5.0000000000000002e-136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 66.3%

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

   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

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(z + x, \frac{x - z}{y}, y\right), 0\right)} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(0 - \frac{{z}^{2}}{y}\right)} + y\right) \]

      4. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(0 + \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} + y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(0 + \color{blue}{-1 \cdot \frac{{z}^{2}}{y}}\right) + y\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0 + \left(-1 \cdot \frac{{z}^{2}}{y} + y\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(0 + \color{blue}{\left(y + -1 \cdot \frac{{z}^{2}}{y}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{0} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      10. mul0-rgt N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot 0} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      11. mul0-lft N/A

          \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(0 \cdot \frac{z}{y}\right)} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      12. metadata-eval N/A

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

      13. distribute-lft1-in N/A

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

      14. associate-*r* N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. distribute-lft1-in N/A

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

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, y - z \cdot \frac{z}{y}, 0\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-136}:\\ \;\;\;\;\left(z + x\right) \cdot \left(\left(x - z\right) \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, y - z \cdot \frac{z}{y}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, y - z \cdot \frac{z}{y}, 0\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 0.5 (- y (* z (/ z y))) 0.0))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 -5e-136)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x (/ x y) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(0.5, (y - (z * (z / y))), 0.0);
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= -5e-136) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 0.5 * fma(x, (x / y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(0.5, Float64(y - Float64(z * Float64(z / y))), 0.0)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -5e-136)
		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

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-136], 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))) < -5.0000000000000002e-136 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(z + x, \frac{x - z}{y}, y\right), 0\right)} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(0 - \frac{{z}^{2}}{y}\right)} + y\right) \]

      4. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(0 + \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} + y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(0 + \color{blue}{-1 \cdot \frac{{z}^{2}}{y}}\right) + y\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(0 + \left(-1 \cdot \frac{{z}^{2}}{y} + y\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(0 + \color{blue}{\left(y + -1 \cdot \frac{{z}^{2}}{y}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \color{blue}{0} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      10. mul0-rgt N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot 0} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      11. mul0-lft N/A

          \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(0 \cdot \frac{z}{y}\right)} + \frac{1}{2} \cdot \left(y + -1 \cdot \frac{{z}^{2}}{y}\right) \]

      12. metadata-eval N/A

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

      13. distribute-lft1-in N/A

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

      14. associate-*r* N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. distribute-lft1-in N/A

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

   8. Simplified 70.3%

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

   if -5.0000000000000002e-136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Simplified 66.3%

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

Alternative 5: 51.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-136}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -5e-136)
   (* z (* z (/ -0.5 y)))
   (* 0.5 (fma x (/ x y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -5e-136) {
		tmp = z * (z * (-0.5 / y));
	} else {
		tmp = 0.5 * fma(x, (x / y), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -5e-136)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y)));
	else
		tmp = Float64(0.5 * fma(x, Float64(x / y), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -5e-136], N[(z * N[(z * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-136

   1. Initial program 76.2%

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

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

      11. +-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{\left({z}^{2} + 0\right)} \cdot \frac{-1}{2}}{y} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\color{blue}{z \cdot z} + 0\right) \cdot \frac{-1}{2}}{y} \]

      13. accelerator-lowering-fma.f64 32.1

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

   5. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, z, 0\right) \cdot -0.5}{y}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 32.3

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

   7. Applied egg-rr 32.3%

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

   if -5.0000000000000002e-136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.3%

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

   3. Taylor expanded in z around 0

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

      1. *-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. Simplified 61.6%

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

Alternative 6: 44.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e+21) (* x (* x (/ 0.5 y))) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+21) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.2d+21) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e+21) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e+21:
		tmp = x * (x * (0.5 / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e+21)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.2e+21)
		tmp = x * (x * (0.5 / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e+21], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e21

   1. Initial program 75.7%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} + 0}}{y \cdot 2} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} + 0}{y \cdot 2} \]

      3. accelerator-lowering-fma.f64 32.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}{y \cdot 2} \]

   5. Simplified 32.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}{y \cdot 2} \]
   6. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot x + 0\right) \cdot \frac{1}{y \cdot 2}} \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 35.7

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

   7. Applied egg-rr 35.7%

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

   if 3.2e21 < y

   1. Initial program 46.4%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 61.5

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

   5. Simplified 61.5%

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

Alternative 7: 33.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

def code(x, y, z):
	return 0.5 * y

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 36.1

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

5. Simplified 36.1%

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

Developer Target 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024196 
(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)))