Average Error: 7.8 → 0.4
Time: 11.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.41133532920946507 \cdot 10^{-19} \lor \neg \left(z \le 2.23617164171117214 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x} + e^{-x}}{2}}{x} \cdot \frac{y}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.41133532920946507 \cdot 10^{-19} \lor \neg \left(z \le 2.23617164171117214 \cdot 10^{-52}\right):\\
\;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{e^{x} + e^{-x}}{2}}{x} \cdot \frac{y}{z}\\

\end{array}
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 (((z <= -3.411335329209465e-19) || !(z <= 2.236171641711172e-52))) {
		temp = (((exp(x) + exp(-x)) * y) / (z * (2.0 * x)));
	} else {
		temp = ((((exp(x) + exp(-x)) / 2.0) / x) * (y / z));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.8
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.411335329209465e-19 or 2.236171641711172e-52 < z

    1. Initial program 11.1

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

    3. Applied cosh-def11.1

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

    4. Applied frac-times11.1

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

    5. Applied associate-/l/0.4

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

    if -3.411335329209465e-19 < z < 2.236171641711172e-52

    1. Initial program 0.3

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

    3. Applied cosh-def0.3

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

    4. Applied frac-times0.3

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

    5. Applied associate-/l/22.3

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

    7. Applied *-commutative22.3

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

    8. Applied times-frac0.4

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

    9. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.41133532920946507 \cdot 10^{-19} \lor \neg \left(z \le 2.23617164171117214 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{x} + e^{-x}}{2}}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))