Average Error: 7.9 → 0.4
Time: 3.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1948241059787096980000 \lor \neg \left(z \le 7.2256649558957228 \cdot 10^{-45}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1948241059787096980000 \lor \neg \left(z \le 7.2256649558957228 \cdot 10^{-45}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r524800 = x;
        double r524801 = cosh(r524800);
        double r524802 = y;
        double r524803 = r524802 / r524800;
        double r524804 = r524801 * r524803;
        double r524805 = z;
        double r524806 = r524804 / r524805;
        return r524806;
}
↓
double f(double x, double y, double z) {
        double r524807 = z;
        double r524808 = -1.948241059787097e+21;
        bool r524809 = r524807 <= r524808;
        double r524810 = 7.225664955895723e-45;
        bool r524811 = r524807 <= r524810;
        double r524812 = !r524811;
        bool r524813 = r524809 || r524812;
        double r524814 = x;
        double r524815 = cosh(r524814);
        double r524816 = y;
        double r524817 = r524814 * r524807;
        double r524818 = r524816 / r524817;
        double r524819 = r524815 * r524818;
        double r524820 = r524816 / r524807;
        double r524821 = r524815 * r524820;
        double r524822 = r524821 / r524814;
        double r524823 = r524813 ? r524819 : r524822;
        return r524823;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -1.948241059787097e+21 or 7.225664955895723e-45 < z

    1. Initial program 11.8

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

    3. Applied div-inv11.9

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

    5. Applied associate-*r/11.9

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

    6. Applied associate-*l/10.1

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

    7. Simplified10.0

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

    9. Applied *-un-lft-identity10.0

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

    10. Applied *-un-lft-identity10.0

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

    11. Applied times-frac10.0

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

    12. Applied times-frac10.0

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

    13. Simplified10.0

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

    14. Simplified0.4

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

    if -1.948241059787097e+21 < z < 7.225664955895723e-45

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    5. Applied associate-*r/0.5

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

    6. Applied associate-*l/0.4

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

    7. Simplified0.3

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

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

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

    10. Applied times-frac0.3

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

    11. Simplified0.3

      \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{z}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1948241059787096980000 \lor \neg \left(z \le 7.2256649558957228 \cdot 10^{-45}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))