Average Error: 7.8 → 0.8
Time: 14.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.766253741311095338476931406909394285838 \cdot 10^{-74} \lor \neg \left(y \le 1.349115509152380621030363919966636831956 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.766253741311095338476931406909394285838 \cdot 10^{-74} \lor \neg \left(y \le 1.349115509152380621030363919966636831956 \cdot 10^{-56}\right):\\
\;\;\;\;\frac{\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r347523 = x;
        double r347524 = cosh(r347523);
        double r347525 = y;
        double r347526 = r347525 / r347523;
        double r347527 = r347524 * r347526;
        double r347528 = z;
        double r347529 = r347527 / r347528;
        return r347529;
}
↓
double f(double x, double y, double z) {
        double r347530 = y;
        double r347531 = -2.7662537413110953e-74;
        bool r347532 = r347530 <= r347531;
        double r347533 = 1.3491155091523806e-56;
        bool r347534 = r347530 <= r347533;
        double r347535 = !r347534;
        bool r347536 = r347532 || r347535;
        double r347537 = 0.5;
        double r347538 = x;
        double r347539 = exp(r347538);
        double r347540 = -r347538;
        double r347541 = exp(r347540);
        double r347542 = r347539 + r347541;
        double r347543 = r347537 * r347542;
        double r347544 = r347530 * r347543;
        double r347545 = z;
        double r347546 = r347544 / r347545;
        double r347547 = r347546 / r347538;
        double r347548 = cosh(r347538);
        double r347549 = r347548 * r347530;
        double r347550 = 1.0;
        double r347551 = r347550 / r347538;
        double r347552 = r347549 * r347551;
        double r347553 = r347552 / r347545;
        double r347554 = r347536 ? r347547 : r347553;
        return r347554;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.7662537413110953e-74 or 1.3491155091523806e-56 < y

    1. Initial program 15.7

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

    3. Applied associate-*r/15.7

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

    4. Applied associate-/l/1.2

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

    5. Taylor expanded around inf 1.2

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

    6. Simplified1.3

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

    if -2.7662537413110953e-74 < y < 1.3491155091523806e-56

    1. Initial program 0.3

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

    3. Applied div-inv0.3

      \[\leadsto \frac{\cosh x \cdot \color{blue}{\left(y \cdot \frac{1}{x}\right)}}{z}\]

    4. Applied associate-*r*0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.766253741311095338476931406909394285838 \cdot 10^{-74} \lor \neg \left(y \le 1.349115509152380621030363919966636831956 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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))