Average Error: 8.0 → 0.5
Time: 16.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -318595888080967303168:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -318595888080967303168:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\
\;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r29378360 = x;
        double r29378361 = cosh(r29378360);
        double r29378362 = y;
        double r29378363 = r29378362 / r29378360;
        double r29378364 = r29378361 * r29378363;
        double r29378365 = z;
        double r29378366 = r29378364 / r29378365;
        return r29378366;
}
↓
double f(double x, double y, double z) {
        double r29378367 = y;
        double r29378368 = -3.185958880809673e+20;
        bool r29378369 = r29378367 <= r29378368;
        double r29378370 = x;
        double r29378371 = cosh(r29378370);
        double r29378372 = r29378371 * r29378367;
        double r29378373 = z;
        double r29378374 = r29378372 / r29378373;
        double r29378375 = r29378374 / r29378370;
        double r29378376 = 5.764064057561761e-61;
        bool r29378377 = r29378367 <= r29378376;
        double r29378378 = 0.5;
        double r29378379 = -r29378370;
        double r29378380 = exp(r29378379);
        double r29378381 = exp(r29378370);
        double r29378382 = r29378380 + r29378381;
        double r29378383 = r29378378 * r29378382;
        double r29378384 = r29378383 * r29378367;
        double r29378385 = r29378384 / r29378370;
        double r29378386 = r29378385 / r29378373;
        double r29378387 = r29378377 ? r29378386 : r29378375;
        double r29378388 = r29378369 ? r29378375 : r29378387;
        return r29378388;
}

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

Original8.0
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.185958880809673e+20 or 5.764064057561761e-61 < y

    1. Initial program 19.3

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

    3. Applied associate-*r/19.3

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

    4. Applied associate-/l/0.7

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
    5. Using strategy rm

    6. Applied associate-/r*0.8

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

    if -3.185958880809673e+20 < y < 5.764064057561761e-61

    1. Initial program 0.4

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

    3. Applied associate-*r/0.4

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

    4. Applied associate-/l/10.6

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
    5. Using strategy rm

    6. Applied associate-/l*11.0

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z \cdot x}{y}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity11.0

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

    9. Applied times-frac0.8

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

    10. Applied associate-/r*0.5

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

    11. Simplified0.5

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

    12. Taylor expanded around inf 10.6

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

    13. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -318595888080967303168:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))