Average Error: 7.8 → 0.5
Time: 13.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -134764098203869446144 \lor \neg \left(y \le 1.481740486521299863421596595564379045588 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)\right) \cdot \frac{y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -134764098203869446144 \lor \neg \left(y \le 1.481740486521299863421596595564379045588 \cdot 10^{-48}\right):\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r368429 = x;
        double r368430 = cosh(r368429);
        double r368431 = y;
        double r368432 = r368431 / r368429;
        double r368433 = r368430 * r368432;
        double r368434 = z;
        double r368435 = r368433 / r368434;
        return r368435;
}
↓
double f(double x, double y, double z) {
        double r368436 = y;
        double r368437 = -1.3476409820386945e+20;
        bool r368438 = r368436 <= r368437;
        double r368439 = 1.4817404865213e-48;
        bool r368440 = r368436 <= r368439;
        double r368441 = !r368440;
        bool r368442 = r368438 || r368441;
        double r368443 = x;
        double r368444 = exp(r368443);
        double r368445 = -r368443;
        double r368446 = exp(r368445);
        double r368447 = r368444 + r368446;
        double r368448 = r368447 * r368436;
        double r368449 = z;
        double r368450 = r368448 / r368449;
        double r368451 = 2.0;
        double r368452 = r368451 * r368443;
        double r368453 = r368450 / r368452;
        double r368454 = 0.5;
        double r368455 = -1.0;
        double r368456 = r368455 * r368443;
        double r368457 = exp(r368456);
        double r368458 = r368457 + r368444;
        double r368459 = r368454 * r368458;
        double r368460 = r368436 / r368443;
        double r368461 = r368459 * r368460;
        double r368462 = r368461 / r368449;
        double r368463 = r368442 ? r368453 : r368462;
        return r368463;
}

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
Target0.5
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 3 regimes

  2. if y < -1.3476409820386945e+20

    1. Initial program 24.7

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

    3. Applied cosh-def24.7

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

    4. Applied frac-times24.7

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

    5. Applied associate-/l/0.4

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

    7. Applied associate-/r*0.4

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

    9. Applied div-inv0.5

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

    10. Applied times-frac0.5

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

    if -1.3476409820386945e+20 < y < 1.4817404865213e-48

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if 1.4817404865213e-48 < y

    1. Initial program 16.9

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

    3. Applied cosh-def17.0

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

    4. Applied frac-times17.0

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

    5. Applied associate-/l/0.7

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

    7. Applied associate-/r*0.8

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

    9. Applied div-inv0.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -134764098203869446144 \lor \neg \left(y \le 1.481740486521299863421596595564379045588 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)\right) \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))