Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.4

Time: 9.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -6.9485940714744041 \cdot 10^{277} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 7.54740556372758867 \cdot 10^{214}\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{\frac{1}{2}}{\frac{z}{e^{-1 \cdot x} + e^{x}}} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if ((((cosh(x) * (y / x)) <= -6.948594071474404e+277) || !((cosh(x) * (y / x)) <= 7.547405563727589e+214))) {
		temp = ((1.0 / x) * ((0.5 / (z / (exp((-1.0 * x)) + exp(x)))) * y));
	} else {
		temp = ((cosh(x) * (y / x)) / z);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	0.5
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* (cosh x) (/ y x)) < -6.948594071474404e+277 or 7.547405563727589e+214 < (* (cosh x) (/ y x))

   1. Initial program 38.7

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.6

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

   5. Applied *-un-lft-identity 0.6

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

   6. Applied times-frac 35.0

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

   7. Applied associate-/r* 35.0

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

   8. Simplified 35.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{\frac{z}{e^{-1 \cdot x} + e^{x}}}}}{\frac{x}{y}}\]
   9. Using strategy rm

   10. Applied div-inv 35.0

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

   11. Applied *-un-lft-identity 35.0

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

   12. Applied *-un-lft-identity 35.0

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

   13. Applied times-frac 35.0

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

   14. Applied *-un-lft-identity 35.0

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

   15. Applied times-frac 35.0

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

   16. Applied times-frac 0.9

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

   17. Simplified 0.9

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

   18. Simplified 0.9

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

   if -6.948594071474404e+277 < (* (cosh x) (/ y x)) < 7.547405563727589e+214

   1. Initial program 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -6.9485940714744041 \cdot 10^{277} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 7.54740556372758867 \cdot 10^{214}\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{\frac{1}{2}}{\frac{z}{e^{-1 \cdot x} + e^{x}}} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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))