Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.4 → 0.3

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -5193.795431700195 \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2.3817947252374567 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z}}{x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ (* (cosh x) (/ y x)) z) -5193.795431700195)
         (not (<= (/ (* (cosh x) (/ y x)) z) 2.3817947252374567e+33)))
   (/ (/ (* y (+ (exp x) (exp (- x)))) z) (* x 2.0))
   (/ (* (cosh x) 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 tmp;
	if ((((cosh(x) * (y / x)) / z) <= -5193.795431700195) || !(((cosh(x) * (y / x)) / z) <= 2.3817947252374567e+33)) {
		tmp = ((y * (exp(x) + exp(-x))) / z) / (x * 2.0);
	} else {
		tmp = (cosh(x) * y) / (x * z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	0.4
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -5193.795431700195 or 2.3817947252374567e33 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 13.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
   2. Using strategy rm

   3. Applied div-inv_binary64 13.2

      \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}}\]
   4. Using strategy rm

   5. Applied cosh-def_binary64 13.2

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

   6. Applied frac-times_binary64 13.2

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

   7. Applied associate-*l/_binary64 0.5

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

   8. Simplified 0.4

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

   if -5193.795431700195 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 2.3817947252374567e33

   1. Initial program 0.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
   2. Using strategy rm

   3. Applied associate-*r/_binary64 0.2

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

   4. Applied associate-/l/_binary64 0.3

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

   5. Simplified 0.3

      \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{x \cdot z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -5193.795431700195 \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2.3817947252374567 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z}}{x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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