Average Error: 7.5 → 0.4
Time: 4.5m
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.7860378529012271 \cdot 10^{-34} \lor \neg \left(z \le 5.32104131336549014 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{1}}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot 2}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.7860378529012271 \cdot 10^{-34} \lor \neg \left(z \le 5.32104131336549014 \cdot 10^{-31}\right):\\
\;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{1}}{z \cdot \left(2 \cdot x\right)}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) (((double) (((double) cosh(x)) * ((double) (y / x)))) / z));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((z <= -4.786037852901227e-34) || !(z <= 5.32104131336549e-31))) {
		VAR = ((double) (((double) (((double) (((double) exp(x)) + ((double) exp(((double) -(x)))))) * ((double) (y / 1.0)))) / ((double) (z * ((double) (2.0 * x))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (((double) exp(x)) + ((double) exp(((double) -(x)))))) * y)) / ((double) (z * 2.0)))) / x));
	}
	return VAR;
}

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

Original7.5
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 < -4.786037852901227e-34 or 5.32104131336549e-31 < z

    1. Initial program 10.5

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

    3. Applied *-un-lft-identity10.5

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

    4. Applied associate-/r*10.5

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

    5. Applied cosh-def10.5

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

    6. Applied frac-times10.5

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

    7. Applied associate-/l/0.4

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

    if -4.786037852901227e-34 < z < 5.32104131336549e-31

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied associate-/r*0.3

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

    5. Applied cosh-def0.3

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

    6. Applied frac-times0.3

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

    7. Applied associate-/l/23.8

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

    9. Applied associate-*r*23.8

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

    10. Applied associate-/r*0.3

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

    11. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.7860378529012271 \cdot 10^{-34} \lor \neg \left(z \le 5.32104131336549014 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{1}}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot 2}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))