Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 31.2% → 35.9%

Time: 15.2s

Precision: binary64

Cost: 26760

Math

\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 480:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\frac{y \cdot x}{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt[3]{z}}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+101)
   (* y (/ 1.0 (* z x)))
   (if (<= y 480.0)
     (fma 0.5 (/ (/ (* y x) (pow (cbrt z) 2.0)) (cbrt z)) (/ (/ y x) z))
     (/ y (* z x)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+101) {
		tmp = y * (1.0 / (z * x));
	} else if (y <= 480.0) {
		tmp = fma(0.5, (((y * x) / pow(cbrt(z), 2.0)) / cbrt(z)), ((y / x) / z));
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+101)
		tmp = Float64(y * Float64(1.0 / Float64(z * x)));
	elseif (y <= 480.0)
		tmp = fma(0.5, Float64(Float64(Float64(y * x) / (cbrt(z) ^ 2.0)) / cbrt(z)), Float64(Float64(y / x) / z));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, -1.7e+101], N[(y * N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 480.0], N[(0.5 * N[(N[(N[(y * x), $MachinePrecision] / N[Power[N[Power[z, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[z, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	31.2%
Target	35.0%
Herbie	35.9%

\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000009e101

   1. Initial program 12.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 23.0%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
      Step-by-step derivation

      [Start] 12.7	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 12.7	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 22.9	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 23.0	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 23.0	\[ \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]
      *-commutative [=>] 23.0	\[ y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   3. Taylor expanded in x around 0 24.4%

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

   if -1.70000000000000009e101 < y < 480

   1. Initial program 42.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 37.1%

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

   3. Simplified 43.7%

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

      [Start] 37.1	\[ \frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z} \]
      +-commutative [=>] 37.1	\[ \color{blue}{0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}} \]
      fma-def [=>] 37.1	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{y \cdot x}{z}, \frac{y}{z \cdot x}\right)} \]
      associate-/l* [=>] 36.6	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{y}{\frac{z}{x}}}, \frac{y}{z \cdot x}\right) \]
      associate-/r/ [=>] 37.1	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{y}{z} \cdot x}, \frac{y}{z \cdot x}\right) \]
      associate-/l/ [<=] 43.7	\[ \mathsf{fma}\left(0.5, \frac{y}{z} \cdot x, \color{blue}{\frac{\frac{y}{x}}{z}}\right) \]

   4. Applied egg-rr 43.7%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y \cdot x}{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt[3]{z}}}, \frac{\frac{y}{x}}{z}\right) \]
      Step-by-step derivation

      [Start] 43.7	\[ \mathsf{fma}\left(0.5, \frac{y}{z} \cdot x, \frac{\frac{y}{x}}{z}\right) \]
      associate-*l/ [=>] 43.7	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{y \cdot x}{z}}, \frac{\frac{y}{x}}{z}\right) \]
      add-cube-cbrt [=>] 43.7	\[ \mathsf{fma}\left(0.5, \frac{y \cdot x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}, \frac{\frac{y}{x}}{z}\right) \]
      associate-/r* [=>] 43.7	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y \cdot x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{\sqrt[3]{z}}}, \frac{\frac{y}{x}}{z}\right) \]
      pow2 [=>] 43.7	\[ \mathsf{fma}\left(0.5, \frac{\frac{y \cdot x}{\color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}}}{\sqrt[3]{z}}, \frac{\frac{y}{x}}{z}\right) \]

   if 480 < y

   1. Initial program 15.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 21.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
      Step-by-step derivation

      [Start] 15.3	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 15.3	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 21.5	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 21.4	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 21.4	\[ \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]
      *-commutative [=>] 21.4	\[ y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   3. Taylor expanded in x around 0 23.4%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 480:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\frac{y \cdot x}{{\left(\sqrt[3]{z}\right)}^{2}}}{\sqrt[3]{z}}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	35.9%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 3400:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{1}{\frac{z}{y \cdot x}}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	35.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+192}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 7800:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{y}{z}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 3
Accuracy	35.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 1950:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 4
Accuracy	35.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \leq 98000:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 5
Accuracy	33.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq 200 \lor \neg \left(y \leq 1.12 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 6
Accuracy	32.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 7
Accuracy	32.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 8
Accuracy	32.0%
Cost	320

\[\frac{y}{z \cdot x} \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))