Average Error: 7.4 → 0.9
Time: 11.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.255320694630306831846011661468764941674 \cdot 10^{-4} \lor \neg \left(z \le 2.126137581571963274216038367069096111718 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x} \cdot \left(\frac{1}{y} \cdot x\right)}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.255320694630306831846011661468764941674 \cdot 10^{-4} \lor \neg \left(z \le 2.126137581571963274216038367069096111718 \cdot 10^{-41}\right):\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{z}{\cosh x} \cdot \left(\frac{1}{y} \cdot x\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r324463 = x;
        double r324464 = cosh(r324463);
        double r324465 = y;
        double r324466 = r324465 / r324463;
        double r324467 = r324464 * r324466;
        double r324468 = z;
        double r324469 = r324467 / r324468;
        return r324469;
}
↓
double f(double x, double y, double z) {
        double r324470 = z;
        double r324471 = -0.0007255320694630307;
        bool r324472 = r324470 <= r324471;
        double r324473 = 2.1261375815719633e-41;
        bool r324474 = r324470 <= r324473;
        double r324475 = !r324474;
        bool r324476 = r324472 || r324475;
        double r324477 = 0.5;
        double r324478 = x;
        double r324479 = y;
        double r324480 = r324478 * r324479;
        double r324481 = r324480 / r324470;
        double r324482 = r324477 * r324481;
        double r324483 = r324478 * r324470;
        double r324484 = r324479 / r324483;
        double r324485 = r324482 + r324484;
        double r324486 = 1.0;
        double r324487 = cosh(r324478);
        double r324488 = r324470 / r324487;
        double r324489 = r324486 / r324479;
        double r324490 = r324489 * r324478;
        double r324491 = r324488 * r324490;
        double r324492 = r324486 / r324491;
        double r324493 = r324476 ? r324485 : r324492;
        return r324493;
}

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

Original7.4
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 z < -0.0007255320694630307 or 2.1261375815719633e-41 < z

    1. Initial program 10.6

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

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

    if -0.0007255320694630307 < z < 2.1261375815719633e-41

    1. Initial program 0.3

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

    3. Applied add-exp-log0.3

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

    5. Applied div-inv0.4

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

    6. Applied associate-*r*0.4

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

    7. Simplified0.4

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

    9. Applied clear-num0.5

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

    10. Simplified0.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{y}}{\frac{\cosh x}{x}}}}\]
    11. Using strategy rm

    12. Applied div-inv0.5

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

    13. Applied div-inv0.6

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

    14. Applied times-frac0.6

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

    15. Simplified0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.255320694630306831846011661468764941674 \cdot 10^{-4} \lor \neg \left(z \le 2.126137581571963274216038367069096111718 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x} \cdot \left(\frac{1}{y} \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(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))