Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.3

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -10000000000:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right)\\ \mathbf{elif}\;y \leq 10^{-13}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{{z}^{-1}}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -10000000000.0)
   (* (/ y (* z x)) (fma 0.5 (exp x) (/ 0.5 (exp x))))
   (if (<= y 1e-13)
     (/ (* (cosh x) (/ y x)) z)
     (* (cosh x) (* y (/ (pow z -1.0) 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 <= -10000000000.0) {
		tmp = (y / (z * x)) * fma(0.5, exp(x), (0.5 / exp(x)));
	} else if (y <= 1e-13) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = cosh(x) * (y * (pow(z, -1.0) / 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 <= -10000000000.0)
		tmp = Float64(Float64(y / Float64(z * x)) * fma(0.5, exp(x), Float64(0.5 / exp(x))));
	elseif (y <= 1e-13)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = Float64(cosh(x) * Float64(y * Float64((z ^ -1.0) / 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, -10000000000.0], N[(N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Exp[x], $MachinePrecision] + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-13], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(N[Power[z, -1.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.9
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1e10

   1. Initial program 23.0

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

   2. Taylor expanded in x around inf 0.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(\frac{1}{e^{x}} + e^{x}\right) \cdot y}{z \cdot x}} \]

   3. Simplified 0.3

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

   if -1e10 < y < 1e-13

   1. Initial program 0.2

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

   if 1e-13 < y

   1. Initial program 19.6

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

   2. Applied egg-rr 18.1

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

   3. Applied egg-rr 0.4

      \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{{z}^{-1}}{x} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000000:\\ \;\;\;\;\frac{y}{z \cdot x} \cdot \mathsf{fma}\left(0.5, e^{x}, \frac{0.5}{e^{x}}\right)\\ \mathbf{elif}\;y \leq 10^{-13}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{{z}^{-1}}{x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022206 
(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))