Average Error: 7.7 → 0.8
Time: 4.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.3891980105264862 \cdot 10^{217}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 7.8168077226517798 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.3891980105264862 \cdot 10^{217}:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 7.8168077226517798 \cdot 10^{108}:\\
\;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r583978 = x;
        double r583979 = cosh(r583978);
        double r583980 = y;
        double r583981 = r583980 / r583978;
        double r583982 = r583979 * r583981;
        double r583983 = z;
        double r583984 = r583982 / r583983;
        return r583984;
}
↓
double f(double x, double y, double z) {
        double r583985 = x;
        double r583986 = cosh(r583985);
        double r583987 = y;
        double r583988 = r583987 / r583985;
        double r583989 = r583986 * r583988;
        double r583990 = -5.389198010526486e+217;
        bool r583991 = r583989 <= r583990;
        double r583992 = z;
        double r583993 = r583987 / r583992;
        double r583994 = exp(r583985);
        double r583995 = 0.5;
        double r583996 = r583995 / r583994;
        double r583997 = fma(r583994, r583995, r583996);
        double r583998 = r583993 * r583997;
        double r583999 = r583998 / r583985;
        double r584000 = 7.81680772265178e+108;
        bool r584001 = r583989 <= r584000;
        double r584002 = sqrt(r583986);
        double r584003 = r584002 * r583988;
        double r584004 = r584002 * r584003;
        double r584005 = r584004 / r583992;
        double r584006 = r583985 * r583992;
        double r584007 = r583987 / r584006;
        double r584008 = r583986 * r584007;
        double r584009 = r584001 ? r584005 : r584008;
        double r584010 = r583991 ? r583999 : r584009;
        return r584010;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (cosh x) (/ y x)) < -5.389198010526486e+217

    1. Initial program 30.5

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

    3. Applied *-un-lft-identity30.5

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac30.4

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

    5. Simplified30.4

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

    6. Simplified0.8

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

    7. Taylor expanded around inf 0.9

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

    8. Simplified0.7

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

    if -5.389198010526486e+217 < (* (cosh x) (/ y x)) < 7.81680772265178e+108

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied associate-*l*0.3

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

    if 7.81680772265178e+108 < (* (cosh x) (/ y x))

    1. Initial program 18.4

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

    3. Applied *-un-lft-identity18.4

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac18.4

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

    5. Simplified18.4

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

    6. Simplified2.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.3891980105264862 \cdot 10^{217}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 7.8168077226517798 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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))