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

Percentage Accurate: 69.0% → 90.5%

Time: 9.2s

Alternatives: 10

Speedup: 0.8×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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: 90.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;y \leq -5.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_0 \cdot 0.5}{\frac{y}{\mathsf{hypot}\left(y, x\right)}}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y, x\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= y -5.3e+67)
     (/ (* t_0 0.5) (/ y (hypot y x)))
     (if (<= y 7.5e+163)
       (/ (fma y y (fma x x (* z (- z)))) (* y 2.0))
       (* t_0 (* 0.5 (/ (hypot y x) y)))))))

C

double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if (y <= -5.3e+67) {
		tmp = (t_0 * 0.5) / (y / hypot(y, x));
	} else if (y <= 7.5e+163) {
		tmp = fma(y, y, fma(x, x, (z * -z))) / (y * 2.0);
	} else {
		tmp = t_0 * (0.5 * (hypot(y, x) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (y <= -5.3e+67)
		tmp = Float64(Float64(t_0 * 0.5) / Float64(y / hypot(y, x)));
	elseif (y <= 7.5e+163)
		tmp = Float64(fma(y, y, fma(x, x, Float64(z * Float64(-z)))) / Float64(y * 2.0));
	else
		tmp = Float64(t_0 * Float64(0.5 * Float64(hypot(y, x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[y, -5.3e+67], N[(N[(t$95$0 * 0.5), $MachinePrecision] / N[(y / N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+163], N[(N[(y * y + N[(x * x + N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 * N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3e67

   1. Initial program 29.5%

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

      1. div-inv 29.5%

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

      2. add-sqr-sqrt 25.3%

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

      3. associate-*l* 25.3%

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

   3. Applied egg-rr 89.3%

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

   4. Taylor expanded in z around 0 30.5%

      \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{y} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 30.5%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{{x}^{2} + {y}^{2}}}{y}}\right) \]

      2. *-lft-identity 30.5%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\color{blue}{\sqrt{{x}^{2} + {y}^{2}}}}{y}\right) \]

      3. +-commutative 30.5%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}{y}\right) \]

      4. unpow2 30.5%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{y \cdot y} + {x}^{2}}}{y}\right) \]

      5. unpow2 30.5%

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

      6. hypot-def 89.3%

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

   6. Simplified 89.3%

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

      1. associate-*r* 89.3%

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

      2. clear-num 89.3%

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

      3. un-div-inv 89.3%

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

   8. Applied egg-rr 89.3%

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

   if -5.3e67 < y < 7.50000000000000001e163

   1. Initial program 91.8%

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

      1. sqr-neg 91.8%

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

      2. sqr-neg 91.8%

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

      3. +-commutative 91.8%

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

      4. associate--l+ 91.8%

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

      5. fma-def 92.4%

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

      6. fma-neg 95.2%

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

      7. distribute-rgt-neg-in 95.2%

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

   3. Simplified 95.2%

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

   if 7.50000000000000001e163 < y

   1. Initial program 8.5%

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

      1. div-inv 8.5%

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

      2. add-sqr-sqrt 8.5%

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

      3. associate-*l* 8.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in z around 0 9.2%

      \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{y} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 9.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{{x}^{2} + {y}^{2}}}{y}}\right) \]

      2. *-lft-identity 9.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\color{blue}{\sqrt{{x}^{2} + {y}^{2}}}}{y}\right) \]

      3. +-commutative 9.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}{y}\right) \]

      4. unpow2 9.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{y \cdot y} + {x}^{2}}}{y}\right) \]

      5. unpow2 9.2%

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

      6. hypot-def 86.6%

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

   6. Simplified 86.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot 0.5}{\frac{y}{\mathsf{hypot}\left(y, x\right)}}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y, x\right)}{y}\right)\\ \end{array} \]

Alternative 2: 90.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+67} \lor \neg \left(y \leq 7.5 \cdot 10^{+163}\right):\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y, x\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.3e+67) (not (<= y 7.5e+163)))
   (* (hypot y (hypot x z)) (* 0.5 (/ (hypot y x) y)))
   (/ (fma y y (fma x x (* z (- z)))) (* y 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.3e+67) || !(y <= 7.5e+163)) {
		tmp = hypot(y, hypot(x, z)) * (0.5 * (hypot(y, x) / y));
	} else {
		tmp = fma(y, y, fma(x, x, (z * -z))) / (y * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.3e+67) || !(y <= 7.5e+163))
		tmp = Float64(hypot(y, hypot(x, z)) * Float64(0.5 * Float64(hypot(y, x) / y)));
	else
		tmp = Float64(fma(y, y, fma(x, x, Float64(z * Float64(-z)))) / Float64(y * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.3e+67], N[Not[LessEqual[y, 7.5e+163]], $MachinePrecision]], N[(N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.5 * N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y + N[(x * x + N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3e67 or 7.50000000000000001e163 < y

   1. Initial program 21.3%

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

      1. div-inv 21.3%

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

      2. add-sqr-sqrt 18.7%

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

      3. associate-*l* 18.7%

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

   3. Applied egg-rr 88.2%

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

   4. Taylor expanded in z around 0 22.2%

      \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{y} \cdot \sqrt{{x}^{2} + {y}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 22.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{{x}^{2} + {y}^{2}}}{y}}\right) \]

      2. *-lft-identity 22.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\color{blue}{\sqrt{{x}^{2} + {y}^{2}}}}{y}\right) \]

      3. +-commutative 22.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}{y}\right) \]

      4. unpow2 22.2%

         \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\sqrt{\color{blue}{y \cdot y} + {x}^{2}}}{y}\right) \]

      5. unpow2 22.2%

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

      6. hypot-def 88.3%

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

   6. Simplified 88.3%

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

   if -5.3e67 < y < 7.50000000000000001e163

   1. Initial program 91.8%

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

      1. sqr-neg 91.8%

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

      2. sqr-neg 91.8%

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

      3. +-commutative 91.8%

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

      4. associate--l+ 91.8%

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

      5. fma-def 92.4%

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

      6. fma-neg 95.2%

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

      7. distribute-rgt-neg-in 95.2%

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

   3. Simplified 95.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+67} \lor \neg \left(y \leq 7.5 \cdot 10^{+163}\right):\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \left(0.5 \cdot \frac{\mathsf{hypot}\left(y, x\right)}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \end{array} \]

Alternative 3: 88.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot -0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= y -1.35e+154)
     (* t_0 -0.5)
     (if (<= y 7.5e+163)
       (/ (fma y y (fma x x (* z (- z)))) (* y 2.0))
       (* t_0 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if (y <= -1.35e+154) {
		tmp = t_0 * -0.5;
	} else if (y <= 7.5e+163) {
		tmp = fma(y, y, fma(x, x, (z * -z))) / (y * 2.0);
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(t_0 * -0.5);
	elseif (y <= 7.5e+163)
		tmp = Float64(fma(y, y, fma(x, x, Float64(z * Float64(-z)))) / Float64(y * 2.0));
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[y, -1.35e+154], N[(t$95$0 * -0.5), $MachinePrecision], If[LessEqual[y, 7.5e+163], N[(N[(y * y + N[(x * x + N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 11.1%

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

      1. div-inv 11.1%

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

      2. add-sqr-sqrt 11.1%

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

      3. associate-*l* 11.1%

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

   3. Applied egg-rr 90.2%

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

   4. Taylor expanded in y around -inf 75.9%

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

   if -1.35000000000000003e154 < y < 7.50000000000000001e163

   1. Initial program 89.6%

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

      1. sqr-neg 89.6%

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

      2. sqr-neg 89.6%

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

      3. +-commutative 89.6%

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

      4. associate--l+ 89.6%

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

      5. fma-def 90.2%

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

      6. fma-neg 92.7%

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

      7. distribute-rgt-neg-in 92.7%

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

   3. Simplified 92.7%

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

   if 7.50000000000000001e163 < y

   1. Initial program 8.5%

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

      1. div-inv 8.5%

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

      2. add-sqr-sqrt 8.5%

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

      3. associate-*l* 8.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y around inf 79.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, z \cdot \left(-z\right)\right)\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot 0.5\\ \end{array} \]

Alternative 4: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right)\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot -0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (hypot y (hypot x z))))
   (if (<= y -1.35e+154)
     (* t_0 -0.5)
     (if (<= y 1.3e+146)
       (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
       (if (<= y 1.55e+170) (/ -0.5 (/ (/ y z) z)) (* t_0 0.5))))))

C

double code(double x, double y, double z) {
	double t_0 = hypot(y, hypot(x, z));
	double tmp;
	if (y <= -1.35e+154) {
		tmp = t_0 * -0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.hypot(y, Math.hypot(x, z));
	double tmp;
	if (y <= -1.35e+154) {
		tmp = t_0 * -0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.hypot(y, math.hypot(x, z))
	tmp = 0
	if y <= -1.35e+154:
		tmp = t_0 * -0.5
	elif y <= 1.3e+146:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	elif y <= 1.55e+170:
		tmp = -0.5 / ((y / z) / z)
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = hypot(y, hypot(x, z))
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(t_0 * -0.5);
	elseif (y <= 1.3e+146)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (y <= 1.55e+170)
		tmp = Float64(-0.5 / Float64(Float64(y / z) / z));
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = hypot(y, hypot(x, z));
	tmp = 0.0;
	if (y <= -1.35e+154)
		tmp = t_0 * -0.5;
	elseif (y <= 1.3e+146)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	elseif (y <= 1.55e+170)
		tmp = -0.5 / ((y / z) / z);
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[y, -1.35e+154], N[(t$95$0 * -0.5), $MachinePrecision], If[LessEqual[y, 1.3e+146], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+170], N[(-0.5 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 11.1%

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

      1. div-inv 11.1%

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

      2. add-sqr-sqrt 11.1%

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

      3. associate-*l* 11.1%

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

   3. Applied egg-rr 90.2%

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

   4. Taylor expanded in y around -inf 75.9%

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

   if -1.35000000000000003e154 < y < 1.30000000000000007e146

   1. Initial program 91.8%

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

   if 1.30000000000000007e146 < y < 1.55e170

   1. Initial program 23.2%

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

   2. Taylor expanded in z around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]

      2. clear-num 33.6%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{{z}^{2}}}} \]

      3. un-div-inv 33.6%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 33.6%

         \[\leadsto \frac{-0.5}{\frac{\color{blue}{1 \cdot y}}{{z}^{2}}} \]

      2. unpow2 33.6%

         \[\leadsto \frac{-0.5}{\frac{1 \cdot y}{\color{blue}{z \cdot z}}} \]

      3. times-frac 75.4%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]

   8. Applied egg-rr 75.4%

      \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.7%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot \frac{y}{z}}{z}}} \]

      2. *-lft-identity 75.7%

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

   10. Simplified 75.7%

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

   if 1.55e170 < y

   1. Initial program 9.3%

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

      1. div-inv 9.3%

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

      2. add-sqr-sqrt 9.3%

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

      3. associate-*l* 9.3%

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

   3. Applied egg-rr 92.4%

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

   4. Taylor expanded in y around inf 84.9%

      \[\leadsto \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot \color{blue}{0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot 0.5\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+154)
   (* (hypot y (hypot x z)) -0.5)
   (if (<= y 1.3e+146)
     (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
     (if (<= y 1.55e+170) (/ -0.5 (/ (/ y z) z)) (* y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = hypot(y, hypot(x, z)) * -0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = Math.hypot(y, Math.hypot(x, z)) * -0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+154:
		tmp = math.hypot(y, math.hypot(x, z)) * -0.5
	elif y <= 1.3e+146:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	elif y <= 1.55e+170:
		tmp = -0.5 / ((y / z) / z)
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(hypot(y, hypot(x, z)) * -0.5);
	elseif (y <= 1.3e+146)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (y <= 1.55e+170)
		tmp = Float64(-0.5 / Float64(Float64(y / z) / z));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+154)
		tmp = hypot(y, hypot(x, z)) * -0.5;
	elseif (y <= 1.3e+146)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	elseif (y <= 1.55e+170)
		tmp = -0.5 / ((y / z) / z);
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+154], N[(N[Sqrt[y ^ 2 + N[Sqrt[x ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 1.3e+146], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+170], N[(-0.5 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.35000000000000003e154

   1. Initial program 11.1%

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

      1. div-inv 11.1%

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

      2. add-sqr-sqrt 11.1%

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

      3. associate-*l* 11.1%

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

   3. Applied egg-rr 90.2%

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

   4. Taylor expanded in y around -inf 75.9%

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

   if -1.35000000000000003e154 < y < 1.30000000000000007e146

   1. Initial program 91.8%

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

   if 1.30000000000000007e146 < y < 1.55e170

   1. Initial program 23.2%

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

   2. Taylor expanded in z around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]

      2. clear-num 33.6%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{{z}^{2}}}} \]

      3. un-div-inv 33.6%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 33.6%

         \[\leadsto \frac{-0.5}{\frac{\color{blue}{1 \cdot y}}{{z}^{2}}} \]

      2. unpow2 33.6%

         \[\leadsto \frac{-0.5}{\frac{1 \cdot y}{\color{blue}{z \cdot z}}} \]

      3. times-frac 75.4%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]

   8. Applied egg-rr 75.4%

      \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.7%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot \frac{y}{z}}{z}}} \]

      2. *-lft-identity 75.7%

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

   10. Simplified 75.7%

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

   if 1.55e170 < y

   1. Initial program 9.3%

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

   2. Taylor expanded in y around inf 84.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{hypot}\left(y, \mathsf{hypot}\left(x, z\right)\right) \cdot -0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.35e+154)
   (* y 0.5)
   (if (<= y 1.3e+146)
     (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))
     (if (<= y 1.55e+170) (/ -0.5 (/ (/ y z) z)) (* y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = y * 0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.35d+154)) then
        tmp = y * 0.5d0
    else if (y <= 1.3d+146) then
        tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
    else if (y <= 1.55d+170) then
        tmp = (-0.5d0) / ((y / z) / z)
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.35e+154) {
		tmp = y * 0.5;
	} else if (y <= 1.3e+146) {
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	} else if (y <= 1.55e+170) {
		tmp = -0.5 / ((y / z) / z);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.35e+154:
		tmp = y * 0.5
	elif y <= 1.3e+146:
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	elif y <= 1.55e+170:
		tmp = -0.5 / ((y / z) / z)
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.35e+154)
		tmp = Float64(y * 0.5);
	elseif (y <= 1.3e+146)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0));
	elseif (y <= 1.55e+170)
		tmp = Float64(-0.5 / Float64(Float64(y / z) / z));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.35e+154)
		tmp = y * 0.5;
	elseif (y <= 1.3e+146)
		tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	elseif (y <= 1.55e+170)
		tmp = -0.5 / ((y / z) / z);
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.35e+154], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 1.3e+146], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+170], N[(-0.5 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35000000000000003e154 or 1.55e170 < y

   1. Initial program 10.3%

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

   2. Taylor expanded in y around inf 79.8%

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

   if -1.35000000000000003e154 < y < 1.30000000000000007e146

   1. Initial program 91.8%

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

   if 1.30000000000000007e146 < y < 1.55e170

   1. Initial program 23.2%

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

   2. Taylor expanded in z around inf 33.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. *-commutative 33.6%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]

      2. clear-num 33.6%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{{z}^{2}}}} \]

      3. un-div-inv 33.6%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 33.6%

         \[\leadsto \frac{-0.5}{\frac{\color{blue}{1 \cdot y}}{{z}^{2}}} \]

      2. unpow2 33.6%

         \[\leadsto \frac{-0.5}{\frac{1 \cdot y}{\color{blue}{z \cdot z}}} \]

      3. times-frac 75.4%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]

   8. Applied egg-rr 75.4%

      \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.7%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot \frac{y}{z}}{z}}} \]

      2. *-lft-identity 75.7%

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

   10. Simplified 75.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+170}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 42.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-122}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-28} \lor \neg \left(x \leq 4.5 \cdot 10^{+24}\right) \land x \leq 4.1 \cdot 10^{+108}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.2e-122)
   (* y 0.5)
   (if (or (<= x 1.2e-28) (and (not (<= x 4.5e+24)) (<= x 4.1e+108)))
     (* -0.5 (* z (/ z y)))
     (* x (* x (/ 0.5 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e-122) {
		tmp = y * 0.5;
	} else if ((x <= 1.2e-28) || (!(x <= 4.5e+24) && (x <= 4.1e+108))) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.2d-122) then
        tmp = y * 0.5d0
    else if ((x <= 1.2d-28) .or. (.not. (x <= 4.5d+24)) .and. (x <= 4.1d+108)) then
        tmp = (-0.5d0) * (z * (z / y))
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.2e-122) {
		tmp = y * 0.5;
	} else if ((x <= 1.2e-28) || (!(x <= 4.5e+24) && (x <= 4.1e+108))) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.2e-122:
		tmp = y * 0.5
	elif (x <= 1.2e-28) or (not (x <= 4.5e+24) and (x <= 4.1e+108)):
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.2e-122)
		tmp = Float64(y * 0.5);
	elseif ((x <= 1.2e-28) || (!(x <= 4.5e+24) && (x <= 4.1e+108)))
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.2e-122)
		tmp = y * 0.5;
	elseif ((x <= 1.2e-28) || (~((x <= 4.5e+24)) && (x <= 4.1e+108)))
		tmp = -0.5 * (z * (z / y));
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.2e-122], N[(y * 0.5), $MachinePrecision], If[Or[LessEqual[x, 1.2e-28], And[N[Not[LessEqual[x, 4.5e+24]], $MachinePrecision], LessEqual[x, 4.1e+108]]], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.2000000000000002e-122

   1. Initial program 71.4%

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

   2. Taylor expanded in y around inf 38.3%

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

   if 3.2000000000000002e-122 < x < 1.2000000000000001e-28 or 4.50000000000000019e24 < x < 4.0999999999999999e108

   1. Initial program 74.4%

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

   2. Taylor expanded in z around inf 46.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 46.3%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. div-inv 46.3%

         \[\leadsto \color{blue}{\left({z}^{2} \cdot \frac{1}{y}\right)} \cdot -0.5 \]

      2. unpow2 46.3%

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{1}{y}\right) \cdot -0.5 \]

      3. associate-*l* 53.8%

         \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)} \cdot -0.5 \]

      4. div-inv 53.8%

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

   6. Applied egg-rr 53.8%

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

   if 1.2000000000000001e-28 < x < 4.50000000000000019e24 or 4.0999999999999999e108 < x

   1. Initial program 63.0%

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

   2. Taylor expanded in x around inf 60.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 60.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 60.5%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 60.5%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      2. unpow2 60.5%

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

      3. associate-*r* 74.7%

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

   6. Applied egg-rr 74.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-122}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-28} \lor \neg \left(x \leq 4.5 \cdot 10^{+24}\right) \land x \leq 4.1 \cdot 10^{+108}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 8: 42.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+24} \lor \neg \left(x \leq 1.15 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.5e-129)
   (* y 0.5)
   (if (<= x 1.5e-27)
     (* -0.5 (* z (/ z y)))
     (if (or (<= x 5.8e+24) (not (<= x 1.15e+108)))
       (* x (* x (/ 0.5 y)))
       (/ -0.5 (/ (/ y z) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e-129) {
		tmp = y * 0.5;
	} else if (x <= 1.5e-27) {
		tmp = -0.5 * (z * (z / y));
	} else if ((x <= 5.8e+24) || !(x <= 1.15e+108)) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = -0.5 / ((y / z) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9.5d-129) then
        tmp = y * 0.5d0
    else if (x <= 1.5d-27) then
        tmp = (-0.5d0) * (z * (z / y))
    else if ((x <= 5.8d+24) .or. (.not. (x <= 1.15d+108))) then
        tmp = x * (x * (0.5d0 / y))
    else
        tmp = (-0.5d0) / ((y / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e-129) {
		tmp = y * 0.5;
	} else if (x <= 1.5e-27) {
		tmp = -0.5 * (z * (z / y));
	} else if ((x <= 5.8e+24) || !(x <= 1.15e+108)) {
		tmp = x * (x * (0.5 / y));
	} else {
		tmp = -0.5 / ((y / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.5e-129:
		tmp = y * 0.5
	elif x <= 1.5e-27:
		tmp = -0.5 * (z * (z / y))
	elif (x <= 5.8e+24) or not (x <= 1.15e+108):
		tmp = x * (x * (0.5 / y))
	else:
		tmp = -0.5 / ((y / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.5e-129)
		tmp = Float64(y * 0.5);
	elseif (x <= 1.5e-27)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	elseif ((x <= 5.8e+24) || !(x <= 1.15e+108))
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	else
		tmp = Float64(-0.5 / Float64(Float64(y / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.5e-129)
		tmp = y * 0.5;
	elseif (x <= 1.5e-27)
		tmp = -0.5 * (z * (z / y));
	elseif ((x <= 5.8e+24) || ~((x <= 1.15e+108)))
		tmp = x * (x * (0.5 / y));
	else
		tmp = -0.5 / ((y / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.5e-129], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, 1.5e-27], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 5.8e+24], N[Not[LessEqual[x, 1.15e+108]], $MachinePrecision]], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 9.5000000000000006e-129

   1. Initial program 71.4%

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

   2. Taylor expanded in y around inf 38.3%

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

   if 9.5000000000000006e-129 < x < 1.5000000000000001e-27

   1. Initial program 78.8%

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

   2. Taylor expanded in z around inf 40.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 40.9%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 40.9%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. div-inv 40.9%

         \[\leadsto \color{blue}{\left({z}^{2} \cdot \frac{1}{y}\right)} \cdot -0.5 \]

      2. unpow2 40.9%

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{1}{y}\right) \cdot -0.5 \]

      3. associate-*l* 48.8%

         \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)} \cdot -0.5 \]

      4. div-inv 48.9%

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

   6. Applied egg-rr 48.9%

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

   if 1.5000000000000001e-27 < x < 5.79999999999999958e24 or 1.1499999999999999e108 < x

   1. Initial program 63.0%

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

   2. Taylor expanded in x around inf 60.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 60.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{y}} \]

      2. associate-/l* 60.5%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]

   4. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{x}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 60.5%

         \[\leadsto \color{blue}{\frac{0.5}{y} \cdot {x}^{2}} \]

      2. unpow2 60.5%

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

      3. associate-*r* 74.7%

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

   6. Applied egg-rr 74.7%

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

   if 5.79999999999999958e24 < x < 1.1499999999999999e108

   1. Initial program 70.6%

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

   2. Taylor expanded in z around inf 51.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 51.0%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. *-commutative 51.0%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]

      2. clear-num 51.0%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{y}{{z}^{2}}}} \]

      3. un-div-inv 51.0%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]

   6. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{y}{{z}^{2}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 51.0%

         \[\leadsto \frac{-0.5}{\frac{\color{blue}{1 \cdot y}}{{z}^{2}}} \]

      2. unpow2 51.0%

         \[\leadsto \frac{-0.5}{\frac{1 \cdot y}{\color{blue}{z \cdot z}}} \]

      3. times-frac 58.1%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]

   8. Applied egg-rr 58.1%

      \[\leadsto \frac{-0.5}{\color{blue}{\frac{1}{z} \cdot \frac{y}{z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 58.2%

         \[\leadsto \frac{-0.5}{\color{blue}{\frac{1 \cdot \frac{y}{z}}{z}}} \]

      2. *-lft-identity 58.2%

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

   10. Simplified 58.2%

       \[\leadsto \frac{-0.5}{\color{blue}{\frac{\frac{y}{z}}{z}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-129}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+24} \lor \neg \left(x \leq 1.15 \cdot 10^{+108}\right):\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \end{array} \]

Alternative 9: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{+51} \lor \neg \left(z \leq 7.5 \cdot 10^{+114}\right) \land z \leq 10^{+133}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.18e+51) (and (not (<= z 7.5e+114)) (<= z 1e+133)))
   (* y 0.5)
   (* -0.5 (* z (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.18e+51) || (!(z <= 7.5e+114) && (z <= 1e+133))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= 1.18d+51) .or. (.not. (z <= 7.5d+114)) .and. (z <= 1d+133)) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.18e+51) || (!(z <= 7.5e+114) && (z <= 1e+133))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1.18e+51) or (not (z <= 7.5e+114) and (z <= 1e+133)):
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1.18e+51) || (!(z <= 7.5e+114) && (z <= 1e+133)))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 1.18e+51) || (~((z <= 7.5e+114)) && (z <= 1e+133)))
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1.18e+51], And[N[Not[LessEqual[z, 7.5e+114]], $MachinePrecision], LessEqual[z, 1e+133]]], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.18e51 or 7.5000000000000001e114 < z < 1e133

   1. Initial program 70.4%

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

   2. Taylor expanded in y around inf 38.0%

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

   if 1.18e51 < z < 7.5000000000000001e114 or 1e133 < z

   1. Initial program 71.7%

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

   2. Taylor expanded in z around inf 65.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{{z}^{2}}{y}} \]
   3. Step-by-step derivation

      1. *-commutative 65.3%

         \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]

   4. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{{z}^{2}}{y} \cdot -0.5} \]
   5. Step-by-step derivation

      1. div-inv 65.3%

         \[\leadsto \color{blue}{\left({z}^{2} \cdot \frac{1}{y}\right)} \cdot -0.5 \]

      2. unpow2 65.3%

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{1}{y}\right) \cdot -0.5 \]

      3. associate-*l* 75.1%

         \[\leadsto \color{blue}{\left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)} \cdot -0.5 \]

      4. div-inv 75.1%

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

   6. Applied egg-rr 75.1%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{+51} \lor \neg \left(z \leq 7.5 \cdot 10^{+114}\right) \land z \leq 10^{+133}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 10: 34.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

2. Taylor expanded in y around inf 33.0%

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

3. Final simplification 33.0%

   \[\leadsto y \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× 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 2023318 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))