Average Error: 7.8 → 0.3
Time: 4.3s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\
\;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r420406 = x;
        double r420407 = cosh(r420406);
        double r420408 = y;
        double r420409 = r420408 / r420406;
        double r420410 = r420407 * r420409;
        double r420411 = z;
        double r420412 = r420410 / r420411;
        return r420412;
}
↓
double f(double x, double y, double z) {
        double r420413 = z;
        double r420414 = -3.383191592887116e-08;
        bool r420415 = r420413 <= r420414;
        double r420416 = 3.033555916031553e-25;
        bool r420417 = r420413 <= r420416;
        double r420418 = !r420417;
        bool r420419 = r420415 || r420418;
        double r420420 = y;
        double r420421 = x;
        double r420422 = exp(r420421);
        double r420423 = 0.5;
        double r420424 = r420423 / r420422;
        double r420425 = fma(r420422, r420423, r420424);
        double r420426 = r420420 * r420425;
        double r420427 = r420421 * r420413;
        double r420428 = r420426 / r420427;
        double r420429 = r420420 / r420413;
        double r420430 = r420429 * r420425;
        double r420431 = r420430 / r420421;
        double r420432 = r420419 ? r420428 : r420431;
        return r420432;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.8
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.383191592887116e-08 or 3.033555916031553e-25 < z

    1. Initial program 11.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified10.7

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

    5. Applied associate-*r/10.6

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
    6. Using strategy rm

    7. Applied associate-*l/10.6

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

    8. Applied associate-/l/0.3

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

    if -3.383191592887116e-08 < z < 3.033555916031553e-25

    1. Initial program 0.3

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

    2. Taylor expanded around inf 20.7

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

    3. Simplified0.4

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

    5. Applied associate-*r/0.3

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))