Average Error: 8.0 → 0.6
Time: 14.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\frac{\frac{y}{2 \cdot z} \cdot \frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, {\left(e^{-x}\right)}^{2} - 1\right)}}{x}\\ \mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x} \cdot \frac{y}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -280311605560231821312:\\
\;\;\;\;\frac{\frac{y}{2 \cdot z} \cdot \frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, {\left(e^{-x}\right)}^{2} - 1\right)}}{x}\\

\mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\
\;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r302795 = x;
        double r302796 = cosh(r302795);
        double r302797 = y;
        double r302798 = r302797 / r302795;
        double r302799 = r302796 * r302798;
        double r302800 = z;
        double r302801 = r302799 / r302800;
        return r302801;
}
↓
double f(double x, double y, double z) {
        double r302802 = y;
        double r302803 = -2.8031160556023182e+20;
        bool r302804 = r302802 <= r302803;
        double r302805 = 2.0;
        double r302806 = z;
        double r302807 = r302805 * r302806;
        double r302808 = r302802 / r302807;
        double r302809 = x;
        double r302810 = exp(r302809);
        double r302811 = 3.0;
        double r302812 = pow(r302810, r302811);
        double r302813 = -r302809;
        double r302814 = exp(r302813);
        double r302815 = pow(r302814, r302811);
        double r302816 = r302812 + r302815;
        double r302817 = pow(r302814, r302805);
        double r302818 = 1.0;
        double r302819 = r302817 - r302818;
        double r302820 = fma(r302810, r302810, r302819);
        double r302821 = r302816 / r302820;
        double r302822 = r302808 * r302821;
        double r302823 = r302822 / r302809;
        double r302824 = 5.764064057561761e-61;
        bool r302825 = r302802 <= r302824;
        double r302826 = cosh(r302809);
        double r302827 = r302802 * r302826;
        double r302828 = r302818 / r302809;
        double r302829 = r302827 * r302828;
        double r302830 = r302829 / r302806;
        double r302831 = 0.5;
        double r302832 = r302810 + r302814;
        double r302833 = r302831 * r302832;
        double r302834 = r302833 / r302809;
        double r302835 = r302802 / r302806;
        double r302836 = r302834 * r302835;
        double r302837 = r302825 ? r302830 : r302836;
        double r302838 = r302804 ? r302823 : r302837;
        return r302838;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original8.0
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.8031160556023182e+20

    1. Initial program 23.4

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

    3. Applied cosh-def23.4

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times23.4

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

    5. Applied associate-/l/0.4

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

    6. Simplified0.5

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

    8. Applied flip3-+0.6

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}} \cdot y}{\left(z \cdot 2\right) \cdot x}\]

    9. Applied associate-*l/0.6

      \[\leadsto \frac{\color{blue}{\frac{\left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right) \cdot y}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}}{\left(z \cdot 2\right) \cdot x}\]

    10. Simplified0.6

      \[\leadsto \frac{\frac{\color{blue}{y \cdot \left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right)}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}{\left(z \cdot 2\right) \cdot x}\]
    11. Using strategy rm

    12. Applied associate-/r*0.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{y \cdot \left({\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}\right)}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}}{z \cdot 2}}{x}}\]

    13. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{y}{2 \cdot z} \cdot \frac{{\left(e^{-x}\right)}^{3} + {\left(e^{x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, {\left(e^{-x}\right)}^{2} - 1\right)}}}{x}\]

    if -2.8031160556023182e+20 < y < 5.764064057561761e-61

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    4. Applied associate-*r*0.5

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

    if 5.764064057561761e-61 < y

    1. Initial program 16.7

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

    3. Applied cosh-def16.7

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times16.7

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

    5. Applied associate-/l/0.9

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

    6. Simplified0.9

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

    8. Applied div-inv1.2

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

    9. Simplified1.1

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

    10. Taylor expanded around inf 0.9

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

    11. Simplified1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\frac{\frac{y}{2 \cdot z} \cdot \frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, {\left(e^{-x}\right)}^{2} - 1\right)}}{x}\\ \mathbf{elif}\;y \le 5.764064057561761129309028881747428957517 \cdot 10^{-61}:\\ \;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))