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

Percentage Accurate: 69.5% → 99.9%

Time: 9.0s

Alternatives: 7

Speedup: 1.3×

• 41.7% 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: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.6%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 36.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := z\_m \cdot \left(\frac{z\_m}{y} \cdot -0.5\right)\\ t_1 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (* (/ z_m y) -0.5)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -2e-77)
     t_0
     (if (<= t_1 2e+138)
       (* y 0.5)
       (if (<= t_1 1e+301)
         (/ (* x_m x_m) (* y 2.0))
         (if (<= t_1 INFINITY) (* y 0.5) t_0))))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= 2e+138) {
		tmp = y * 0.5;
	} else if (t_1 <= 1e+301) {
		tmp = (x_m * x_m) / (y * 2.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= 2e+138) {
		tmp = y * 0.5;
	} else if (t_1 <= 1e+301) {
		tmp = (x_m * x_m) / (y * 2.0);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	t_0 = z_m * ((z_m / y) * -0.5)
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -2e-77:
		tmp = t_0
	elif t_1 <= 2e+138:
		tmp = y * 0.5
	elif t_1 <= 1e+301:
		tmp = (x_m * x_m) / (y * 2.0)
	elif t_1 <= math.inf:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(z_m * Float64(Float64(z_m / y) * -0.5))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= 2e+138)
		tmp = Float64(y * 0.5);
	elseif (t_1 <= 1e+301)
		tmp = Float64(Float64(x_m * x_m) / Float64(y * 2.0));
	elseif (t_1 <= Inf)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
x_m = abs(x);
function tmp_2 = code(x_m, y, z_m)
	t_0 = z_m * ((z_m / y) * -0.5);
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= 2e+138)
		tmp = y * 0.5;
	elseif (t_1 <= 1e+301)
		tmp = (x_m * x_m) / (y * 2.0);
	elseif (t_1 <= Inf)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $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, -2e-77], t$95$0, If[LessEqual[t$95$1, 2e+138], N[(y * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y * 2.0), $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))) < -1.9999999999999999e-77 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.9%

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

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -1.9999999999999999e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e138 or 1.00000000000000005e301 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 43.3

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

   5. Applied rewrites 43.3%

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

   if 2.0000000000000001e138 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+301}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := z\_m \cdot \left(\frac{z\_m}{y} \cdot -0.5\right)\\ t_1 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (* (/ z_m y) -0.5)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -2e-77)
     t_0
     (if (<= t_1 2e+138)
       (* y 0.5)
       (if (<= t_1 1e+301)
         (* (* x_m x_m) (/ 0.5 y))
         (if (<= t_1 INFINITY) (* y 0.5) t_0))))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= 2e+138) {
		tmp = y * 0.5;
	} else if (t_1 <= 1e+301) {
		tmp = (x_m * x_m) * (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);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= 2e+138) {
		tmp = y * 0.5;
	} else if (t_1 <= 1e+301) {
		tmp = (x_m * x_m) * (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)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	t_0 = z_m * ((z_m / y) * -0.5)
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -2e-77:
		tmp = t_0
	elif t_1 <= 2e+138:
		tmp = y * 0.5
	elif t_1 <= 1e+301:
		tmp = (x_m * x_m) * (0.5 / y)
	elif t_1 <= math.inf:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(z_m * Float64(Float64(z_m / y) * -0.5))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= 2e+138)
		tmp = Float64(y * 0.5);
	elseif (t_1 <= 1e+301)
		tmp = Float64(Float64(x_m * x_m) * 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);
x_m = abs(x);
function tmp_2 = code(x_m, y, z_m)
	t_0 = z_m * ((z_m / y) * -0.5);
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= 2e+138)
		tmp = y * 0.5;
	elseif (t_1 <= 1e+301)
		tmp = (x_m * x_m) * (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]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $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, -2e-77], t$95$0, If[LessEqual[t$95$1, 2e+138], N[(y * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(0.5 / y), $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))) < -1.9999999999999999e-77 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.9%

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

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -1.9999999999999999e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e138 or 1.00000000000000005e301 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 43.3

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

   5. Applied rewrites 43.3%

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

   if 2.0000000000000001e138 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 54.0

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

   7. Applied rewrites 54.0%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+138}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+301}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (fma (- z_m) (/ z_m y) y)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 INFINITY) (* 0.5 (fma x_m (/ x_m y) y)) t_0))))

C

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

Julia

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

Wolfram

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

   1. Initial program 61.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. *-rgt-identity N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 70.2%

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

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

   1. Initial program 75.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      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 67.4%

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

4. Final simplification 69.0%

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

Alternative 5: 35.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := z\_m \cdot \left(\frac{z\_m}{y} \cdot -0.5\right)\\ t_1 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (* z_m (* (/ z_m y) -0.5)))
        (t_1 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -2e-77) t_0 (if (<= t_1 INFINITY) (* y 0.5) t_0))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
public static double code(double x_m, double y, double z_m) {
	double t_0 = z_m * ((z_m / y) * -0.5);
	double t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -2e-77) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
def code(x_m, y, z_m):
	t_0 = z_m * ((z_m / y) * -0.5)
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -2e-77:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	t_0 = Float64(z_m * Float64(Float64(z_m / y) * -0.5))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
x_m = abs(x);
function tmp_2 = code(x_m, y, z_m)
	t_0 = z_m * ((z_m / y) * -0.5);
	t_1 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -2e-77)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $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, -2e-77], t$95$0, If[LessEqual[t$95$1, Infinity], N[(y * 0.5), $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))) < -1.9999999999999999e-77 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.9%

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

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -1.9999999999999999e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 73.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 39.0

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

   5. Applied rewrites 39.0%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;z\_m \cdot \left(\frac{z\_m}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x\_m, \frac{x\_m}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0)) -2e-77)
   (* z_m (* (/ z_m y) -0.5))
   (* 0.5 (fma x_m (/ x_m y) y))))

C

z_m = fabs(z);
x_m = fabs(x);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-77) {
		tmp = z_m * ((z_m / y) * -0.5);
	} else {
		tmp = 0.5 * fma(x_m, (x_m / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
x_m = abs(x)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -2e-77)
		tmp = Float64(z_m * Float64(Float64(z_m / y) * -0.5));
	else
		tmp = Float64(0.5 * fma(x_m, Float64(x_m / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-77], N[(z$95$m * N[(N[(z$95$m / y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x$95$m * N[(x$95$m / 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))) < -1.9999999999999999e-77

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0

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

   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 z around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 35.3

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

   8. Applied rewrites 35.3%

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

   if -1.9999999999999999e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 58.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      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 63.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;z \cdot \left(\frac{z}{y} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x, \frac{x}{y}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 35.2

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

5. Applied rewrites 35.2%

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

6. Final simplification 35.2%

   \[\leadsto y \cdot 0.5 \]
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 2024233 
(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)))