Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 1.3

Time: 5.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}{x \cdot z}\\ \mathbf{elif}\;y \leq 2.506332563062992 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x \cdot z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+32)
   (* 0.5 (/ (* y (+ (exp x) (/ 1.0 (exp x)))) (* x z)))
   (if (<= y 2.506332563062992e-147)
     (/ (* (cosh x) (/ y x)) z)
     (* 0.5 (/ (fma y (exp x) (/ y (exp x))) (* 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 tmp;
	if (y <= -1e+32) {
		tmp = 0.5 * ((y * (exp(x) + (1.0 / exp(x)))) / (x * z));
	} else if (y <= 2.506332563062992e-147) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = 0.5 * (fma(y, exp(x), (y / exp(x))) / (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)
	tmp = 0.0
	if (y <= -1e+32)
		tmp = Float64(0.5 * Float64(Float64(y * Float64(exp(x) + Float64(1.0 / exp(x)))) / Float64(x * z)));
	elseif (y <= 2.506332563062992e-147)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = Float64(0.5 * Float64(fma(y, exp(x), Float64(y / exp(x))) / 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_] := If[LessEqual[y, -1e+32], N[(0.5 * N[(N[(y * N[(N[Exp[x], $MachinePrecision] + N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.506332563062992e-147], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(0.5 * N[(N[(y * N[Exp[x], $MachinePrecision] + N[(y / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.8
Target	0.4
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000005e32

   1. Initial program 24.1

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

   2. Taylor expanded in x around inf 0.4

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

   if -1.00000000000000005e32 < y < 2.5063325630629919e-147

   1. Initial program 0.3

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

   if 2.5063325630629919e-147 < y

   1. Initial program 12.4

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

   2. Taylor expanded in x around inf 3.2

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

   3. Applied egg-rr 3.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{2 \cdot \cosh x}{x} \cdot \frac{y}{z}\right)} \]

   4. Taylor expanded in x around inf 3.2

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

   5. Simplified 3.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}{x \cdot z}\\ \mathbf{elif}\;y \leq 2.506332563062992 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x \cdot z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))