Average Error: 7.9 → 0.9
Time: 13.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\
\;\;\;\;\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot \frac{y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r424748 = x;
        double r424749 = cosh(r424748);
        double r424750 = y;
        double r424751 = r424750 / r424748;
        double r424752 = r424749 * r424751;
        double r424753 = z;
        double r424754 = r424752 / r424753;
        return r424754;
}
↓
double f(double x, double y, double z) {
        double r424755 = y;
        double r424756 = -9.805921840000082e+80;
        bool r424757 = r424755 <= r424756;
        double r424758 = 1.9092807616058557e-68;
        bool r424759 = r424755 <= r424758;
        double r424760 = !r424759;
        bool r424761 = r424757 || r424760;
        double r424762 = 0.5;
        double r424763 = x;
        double r424764 = -r424763;
        double r424765 = exp(r424764);
        double r424766 = exp(r424763);
        double r424767 = r424765 + r424766;
        double r424768 = r424762 * r424767;
        double r424769 = z;
        double r424770 = r424763 * r424769;
        double r424771 = r424755 / r424770;
        double r424772 = r424768 * r424771;
        double r424773 = r424763 / r424755;
        double r424774 = r424768 / r424773;
        double r424775 = r424774 / r424769;
        double r424776 = r424761 ? r424772 : r424775;
        return r424776;
}

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

Original7.9
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.805921840000082e+80 or 1.9092807616058557e-68 < y

    1. Initial program 20.6

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

    2. Taylor expanded around inf 0.8

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

    3. Simplified0.9

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

    5. Applied associate-/r/0.9

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

    7. Applied div-inv0.9

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

    8. Applied associate-*l*0.9

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

    9. Simplified0.8

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

    if -9.805921840000082e+80 < y < 1.9092807616058557e-68

    1. Initial program 0.8

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

    2. Taylor expanded around inf 0.8

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

    3. Simplified0.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))