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

Percentage Accurate: 69.0% → 99.9%

Time: 7.0s

Alternatives: 8

Speedup: 1.3×

• 42.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.0%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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. Applied rewrites 99.9%

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

Alternative 2: 39.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+137}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* (* -0.5 (/ z_m y)) z_m))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -1e-94)
     t_0
     (if (<= t_1 5e+137)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* (* (/ x y) x) 0.5) t_0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (-0.5 * (z_m / y)) * z_m;
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -1e-94) {
		tmp = t_0;
	} else if (t_1 <= 5e+137) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (-0.5 * (z_m / y)) * z_m;
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -1e-94) {
		tmp = t_0;
	} else if (t_1 <= 5e+137) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (-0.5 * (z_m / y)) * z_m
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -1e-94:
		tmp = t_0
	elif t_1 <= 5e+137:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = ((x / y) * x) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-94)
		tmp = t_0;
	elseif (t_1 <= 5e+137)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (-0.5 * (z_m / y)) * z_m;
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -1e-94)
		tmp = t_0;
	elseif (t_1 <= 5e+137)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = ((x / y) * x) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-94], t$95$0, If[LessEqual[t$95$1, 5e+137], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 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))) < -9.9999999999999996e-95 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.5%

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

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.2%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e137

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if 5.0000000000000002e137 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 44.0%

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

Alternative 3: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m}{\frac{y}{z\_m}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m + x\right) \cdot 0.5\right) \cdot \frac{x - z\_m}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -1e-94)
     (* -0.5 (/ z_m (/ y z_m)))
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (* (+ z_m x) 0.5) (/ (- x z_m) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-94) {
		tmp = -0.5 * (z_m / (y / z_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = ((z_m + x) * 0.5) * ((x - z_m) / y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-94)
		tmp = Float64(-0.5 * Float64(z_m / Float64(y / z_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	else
		tmp = Float64(Float64(Float64(z_m + x) * 0.5) * Float64(Float64(x - z_m) / y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-94], N[(-0.5 * N[(z$95$m / N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -9.9999999999999996e-95

   1. Initial program 83.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 33.2%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 75.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. difference-of-squares N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 4: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m}{\frac{y}{z\_m}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z\_m, \frac{z\_m}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -1e-94)
     (* -0.5 (/ z_m (/ y z_m)))
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (fma (- z_m) (/ z_m y) y) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-94) {
		tmp = -0.5 * (z_m / (y / z_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = fma(-z_m, (z_m / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-94)
		tmp = Float64(-0.5 * Float64(z_m / Float64(y / z_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	else
		tmp = Float64(fma(Float64(-z_m), Float64(z_m / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-94], N[(-0.5 * N[(z$95$m / N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-z$95$m) * N[(z$95$m / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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))) < -9.9999999999999996e-95

   1. Initial program 83.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 33.2%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 75.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   7. Applied rewrites 62.1%

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

Alternative 5: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z\_m, \frac{z\_m}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -1e-94)
     (* (* -0.5 (/ z_m y)) z_m)
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (fma (- z_m) (/ z_m y) y) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -1e-94) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = fma(-z_m, (z_m / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-94)
		tmp = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	else
		tmp = Float64(fma(Float64(-z_m), Float64(z_m / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-94], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[((-z$95$m) * N[(z$95$m / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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))) < -9.9999999999999996e-95

   1. Initial program 83.5%

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

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 33.1%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 75.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 48.8

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

   5. Applied rewrites 48.8%

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

   7. Applied rewrites 62.1%

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

Alternative 6: 35.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-94} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (or (<= t_0 -1e-94) (not (<= t_0 INFINITY)))
     (* (* -0.5 (/ z_m y)) z_m)
     (* 0.5 y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if ((t_0 <= -1e-94) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if ((t_0 <= -1e-94) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if (t_0 <= -1e-94) or not (t_0 <= math.inf):
		tmp = (-0.5 * (z_m / y)) * z_m
	else:
		tmp = 0.5 * y
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if ((t_0 <= -1e-94) || !(t_0 <= Inf))
		tmp = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if ((t_0 <= -1e-94) || ~((t_0 <= Inf)))
		tmp = (-0.5 * (z_m / y)) * z_m;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-94], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(0.5 * y), $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))) < -9.9999999999999996e-95 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.5%

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

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.2%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-94} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;\left(-0.5 \cdot \frac{z}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -1 \cdot 10^{-94}:\\ \;\;\;\;\left(-0.5 \cdot \frac{z\_m}{y}\right) \cdot z\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -1e-94)
   (* (* -0.5 (/ z_m y)) z_m)
   (* (fma (/ x y) x y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -1e-94) {
		tmp = (-0.5 * (z_m / y)) * z_m;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -1e-94)
		tmp = Float64(Float64(-0.5 * Float64(z_m / y)) * z_m);
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -1e-94], N[(N[(-0.5 * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $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))) < -9.9999999999999996e-95

   1. Initial program 83.5%

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

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

      2. div-sub N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 33.1%

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

   if -9.9999999999999996e-95 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. *-inverses N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

   5. Applied rewrites 68.8%

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

Alternative 8: 33.6% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 31.7

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

5. Applied rewrites 31.7%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024307 
(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)))