Average Error: 7.6 → 0.7
Time: 3.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 3.7391398403450612 \cdot 10^{37}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{z \cdot x}}{\frac{1}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

\mathbf{elif}\;z \le 3.7391398403450612 \cdot 10^{37}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cosh x}{z \cdot x}}{\frac{1}{y}}\\

\end{array}
double f(double x, double y, double z) {
        double r540694 = x;
        double r540695 = cosh(r540694);
        double r540696 = y;
        double r540697 = r540696 / r540694;
        double r540698 = r540695 * r540697;
        double r540699 = z;
        double r540700 = r540698 / r540699;
        return r540700;
}
↓
double f(double x, double y, double z) {
        double r540701 = z;
        double r540702 = -1.3159694493115124e+88;
        bool r540703 = r540701 <= r540702;
        double r540704 = x;
        double r540705 = cosh(r540704);
        double r540706 = y;
        double r540707 = r540705 * r540706;
        double r540708 = r540701 * r540704;
        double r540709 = r540707 / r540708;
        double r540710 = 3.739139840345061e+37;
        bool r540711 = r540701 <= r540710;
        double r540712 = r540707 / r540701;
        double r540713 = r540712 / r540704;
        double r540714 = r540705 / r540708;
        double r540715 = 1.0;
        double r540716 = r540715 / r540706;
        double r540717 = r540714 / r540716;
        double r540718 = r540711 ? r540713 : r540717;
        double r540719 = r540703 ? r540709 : r540718;
        return r540719;
}

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

Original7.6
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3159694493115124e+88

    1. Initial program 13.4

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

    3. Applied associate-*r/13.4

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

    4. Applied associate-/l/0.4

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

    if -1.3159694493115124e+88 < z < 3.739139840345061e+37

    1. Initial program 1.5

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

    3. Applied associate-*r/1.5

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

    4. Applied associate-/l/14.2

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

    6. Applied associate-/r*1.1

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

    if 3.739139840345061e+37 < z

    1. Initial program 12.8

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

    3. Applied associate-*r/12.8

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/l*0.8

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

    8. Applied div-inv0.9

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

    9. Applied associate-/r*0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 3.7391398403450612 \cdot 10^{37}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x}{z \cdot x}}{\frac{1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))