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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.3352148123811325 \cdot 10^{50} \lor \neg \left(z \le 1397296929.3794756\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \le -1.3352148123811325 \cdot 10^{50} \lor \neg \left(z \le 1397296929.3794756\right):\\
\;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\

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

\end{array}

double f(double x, double y, double z) {
        double r323607 = x;
        double r323608 = cosh(r323607);
        double r323609 = y;
        double r323610 = r323609 / r323607;
        double r323611 = r323608 * r323610;
        double r323612 = z;
        double r323613 = r323611 / r323612;
        return r323613;
}

↓

double f(double x, double y, double z) {
        double r323614 = z;
        double r323615 = -1.3352148123811325e+50;
        bool r323616 = r323614 <= r323615;
        double r323617 = 1397296929.3794756;
        bool r323618 = r323614 <= r323617;
        double r323619 = !r323618;
        bool r323620 = r323616 || r323619;
        double r323621 = x;
        double r323622 = exp(r323621);
        double r323623 = -r323621;
        double r323624 = exp(r323623);
        double r323625 = r323622 + r323624;
        double r323626 = y;
        double r323627 = r323625 * r323626;
        double r323628 = 2.0;
        double r323629 = r323628 * r323621;
        double r323630 = r323614 * r323629;
        double r323631 = r323627 / r323630;
        double r323632 = r323627 / r323614;
        double r323633 = r323632 / r323629;
        double r323634 = r323620 ? r323631 : r323633;
        return r323634;
}

Original	8.0
Target	0.4
Herbie	0.4

Derivation

Split input into 2 regimes
if z < -1.3352148123811325e+50 or 1397296929.3794756 < z
1. Initial program 13.1
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Using strategy rm
3. Applied cosh-def13.1
  \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]
4. Applied frac-times13.1
  \[\leadsto \frac{\color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2 \cdot x}}}{z}\]
5. Applied associate-/l/0.3
  \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}}\]
if -1.3352148123811325e+50 < z < 1397296929.3794756
1. Initial program 0.6
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Using strategy rm
3. Applied cosh-def0.6
  \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]
4. Applied frac-times0.6
  \[\leadsto \frac{\color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2 \cdot x}}}{z}\]
5. Applied associate-/l/17.3
  \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}}\]
6. Using strategy rm
7. Applied associate-/r*0.5
  \[\leadsto \color{blue}{\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}}\]
Recombined 2 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.3352148123811325 \cdot 10^{50} \lor \neg \left(z \le 1397296929.3794756\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :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))

Error

Try it out

Target

Derivation

`if z < -1.3352148123811325e+50 or 1397296929.3794756 < z`

`if -1.3352148123811325e+50 < z < 1397296929.3794756`

Reproduce

In
Out