Average Error: 7.3 → 0.4
Time: 17.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.752728493811186 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{\left(x \cdot 2\right) \cdot z}\\ \mathbf{elif}\;z \le 2.180956969076489 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(e^{x}, y, \frac{y}{e^{x}}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{\left(x \cdot 2\right) \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.752728493811186 \cdot 10^{+20}:\\
\;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{\left(x \cdot 2\right) \cdot z}\\

\mathbf{elif}\;z \le 2.180956969076489 \cdot 10^{-33}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(e^{x}, y, \frac{y}{e^{x}}\right)}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r25069949 = x;
        double r25069950 = cosh(r25069949);
        double r25069951 = y;
        double r25069952 = r25069951 / r25069949;
        double r25069953 = r25069950 * r25069952;
        double r25069954 = z;
        double r25069955 = r25069953 / r25069954;
        return r25069955;
}
↓
double f(double x, double y, double z) {
        double r25069956 = z;
        double r25069957 = -1.752728493811186e+20;
        bool r25069958 = r25069956 <= r25069957;
        double r25069959 = x;
        double r25069960 = exp(r25069959);
        double r25069961 = 1.0;
        double r25069962 = r25069961 / r25069960;
        double r25069963 = r25069960 + r25069962;
        double r25069964 = y;
        double r25069965 = r25069963 * r25069964;
        double r25069966 = 2.0;
        double r25069967 = r25069959 * r25069966;
        double r25069968 = r25069967 * r25069956;
        double r25069969 = r25069965 / r25069968;
        double r25069970 = 2.180956969076489e-33;
        bool r25069971 = r25069956 <= r25069970;
        double r25069972 = 0.5;
        double r25069973 = r25069964 / r25069960;
        double r25069974 = fma(r25069960, r25069964, r25069973);
        double r25069975 = r25069972 * r25069974;
        double r25069976 = r25069975 / r25069959;
        double r25069977 = r25069976 / r25069956;
        double r25069978 = r25069971 ? r25069977 : r25069969;
        double r25069979 = r25069958 ? r25069969 : r25069978;
        return r25069979;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.3
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.752728493811186e+20 or 2.180956969076489e-33 < z

    1. Initial program 11.1

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

    3. Applied div-inv11.1

      \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}}\]
    4. Using strategy rm

    5. Applied cosh-def11.1

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

    6. Applied frac-times11.1

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

    7. Applied frac-times0.4

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

    8. Simplified0.4

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

    if -1.752728493811186e+20 < z < 2.180956969076489e-33

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied associate-/r*0.3

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

    5. Simplified0.4

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

    7. Applied div-inv0.5

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

    8. Applied *-un-lft-identity0.5

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

    9. Applied times-frac0.5

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

    10. Simplified0.4

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

    11. Taylor expanded around inf 0.3

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

    12. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.752728493811186 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{\left(x \cdot 2\right) \cdot z}\\ \mathbf{elif}\;z \le 2.180956969076489 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(e^{x}, y, \frac{y}{e^{x}}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{\left(x \cdot 2\right) \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))