Average Error: 7.8 → 0.3
Time: 21.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 521563463103.41900634765625:\\ \;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{elif}\;y \le 521563463103.41900634765625:\\
\;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r299583 = x;
        double r299584 = cosh(r299583);
        double r299585 = y;
        double r299586 = r299585 / r299583;
        double r299587 = r299584 * r299586;
        double r299588 = z;
        double r299589 = r299587 / r299588;
        return r299589;
}
↓
double f(double x, double y, double z) {
        double r299590 = y;
        double r299591 = -1.7498766746366753e-11;
        bool r299592 = r299590 <= r299591;
        double r299593 = x;
        double r299594 = cosh(r299593);
        double r299595 = r299594 * r299590;
        double r299596 = z;
        double r299597 = r299595 / r299596;
        double r299598 = r299597 / r299593;
        double r299599 = 521563463103.419;
        bool r299600 = r299590 <= r299599;
        double r299601 = exp(r299593);
        double r299602 = -r299593;
        double r299603 = exp(r299602);
        double r299604 = r299601 + r299603;
        double r299605 = 0.5;
        double r299606 = r299604 * r299605;
        double r299607 = r299590 / r299593;
        double r299608 = r299606 * r299607;
        double r299609 = r299608 / r299596;
        double r299610 = r299596 * r299593;
        double r299611 = r299610 / r299590;
        double r299612 = r299594 / r299611;
        double r299613 = r299600 ? r299609 : r299612;
        double r299614 = r299592 ? r299598 : r299613;
        return r299614;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7498766746366753e-11

    1. Initial program 20.4

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

    3. Applied associate-*r/20.4

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/r*0.3

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

    if -1.7498766746366753e-11 < y < 521563463103.419

    1. Initial program 0.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if 521563463103.419 < y

    1. Initial program 23.6

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

    3. Applied associate-*r/23.6

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

    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*0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 521563463103.41900634765625:\\ \;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}\right) \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))