Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 31.7% → 36.8%

Time: 11.4s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 10^{+247}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{y}{z}, \frac{y}{x \cdot z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 -2e+147)
     (* y (/ (/ 1.0 z) x))
     (if (<= t_0 1e+247) (/ t_0 z) (fma 0.5 (* x (/ y z)) (/ y (* x z)))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= -2e+147) {
		tmp = y * ((1.0 / z) / x);
	} else if (t_0 <= 1e+247) {
		tmp = t_0 / z;
	} else {
		tmp = fma(0.5, (x * (y / z)), (y / (x * z)));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= -2e+147)
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	elseif (t_0 <= 1e+247)
		tmp = Float64(t_0 / z);
	else
		tmp = fma(0.5, Float64(x * Float64(y / z)), Float64(y / Float64(x * z)));
	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_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+147], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+247], N[(t$95$0 / z), $MachinePrecision], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	31.7%
Target	35.6%
Herbie	36.8%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -2e147

   1. Initial program 13.4%

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

   2. Simplified 18.9%

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

      [Start] 13.4	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 13.4	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 19.0	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 18.9	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 18.9	\[ \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]
      *-commutative [=>] 18.9	\[ y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   3. Taylor expanded in x around 0 20.4%

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

   4. Applied egg-rr 20.3%

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

      [Start] 20.4	\[ \frac{y}{z \cdot x} \]
      clear-num [=>] 20.4	\[ \color{blue}{\frac{1}{\frac{z \cdot x}{y}}} \]
      associate-/r/ [=>] 20.3	\[ \color{blue}{\frac{1}{z \cdot x} \cdot y} \]
      associate-/r* [=>] 20.3	\[ \color{blue}{\frac{\frac{1}{z}}{x}} \cdot y \]

   if -2e147 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.99999999999999952e246

   1. Initial program 90.0%

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

   if 9.99999999999999952e246 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 3.5%

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

   2. Simplified 8.6%

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

      [Start] 3.5	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 3.5	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 8.7	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 8.6	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 8.6	\[ \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]
      *-commutative [=>] 8.6	\[ y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   3. Taylor expanded in x around 0 10.5%

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

   4. Simplified 10.8%

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

      [Start] 10.5	\[ \frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z} \]
      +-commutative [=>] 10.5	\[ \color{blue}{0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}} \]
      fma-def [=>] 10.5	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{y \cdot x}{z}, \frac{y}{z \cdot x}\right)} \]
      associate-/l* [=>] 10.4	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{y}{\frac{z}{x}}}, \frac{y}{z \cdot x}\right) \]
      associate-/r/ [=>] 10.8	\[ \mathsf{fma}\left(0.5, \color{blue}{\frac{y}{z} \cdot x}, \frac{y}{z \cdot x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -2 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 10^{+247}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{y}{z}, \frac{y}{x \cdot z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	36.7%
Cost	20424

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 10^{+247}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 2
Accuracy	33.8%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-118}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Accuracy	36.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 4
Accuracy	36.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-67} \lor \neg \left(z \leq 7.5 \cdot 10^{-6}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 5
Accuracy	36.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-67} \lor \neg \left(z \leq 8.2 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 6
Accuracy	32.5%
Cost	320

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

Alternative 7
Accuracy	32.5%
Cost	320

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

Reproduce

herbie shell --seed 2023153 
(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))