Average Error: 7.6 → 2.1
Time: 26.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r456370 = x;
        double r456371 = cosh(r456370);
        double r456372 = y;
        double r456373 = r456372 / r456370;
        double r456374 = r456371 * r456373;
        double r456375 = z;
        double r456376 = r456374 / r456375;
        return r456376;
}
↓
double f(double x, double y, double z) {
        double r456377 = z;
        double r456378 = -1.7216049645256126e+202;
        bool r456379 = r456377 <= r456378;
        double r456380 = 1.3422271353685726e-58;
        bool r456381 = r456377 <= r456380;
        double r456382 = !r456381;
        bool r456383 = r456379 || r456382;
        double r456384 = x;
        double r456385 = cosh(r456384);
        double r456386 = y;
        double r456387 = r456377 * r456384;
        double r456388 = r456386 / r456387;
        double r456389 = r456385 * r456388;
        double r456390 = r456386 / r456377;
        double r456391 = r456390 / r456384;
        double r456392 = r456385 * r456391;
        double r456393 = r456383 ? r456389 : r456392;
        return r456393;
}

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

Original7.6
Target0.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.7216049645256126e+202 or 1.3422271353685726e-58 < z

    1. Initial program 11.7

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

    3. Applied associate-/l*12.2

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

    4. Simplified10.8

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

    6. Applied div-inv10.8

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

    7. Simplified10.7

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

    9. Applied div-inv10.8

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

    10. Applied associate-/l*0.6

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

    11. Simplified0.5

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

    if -1.7216049645256126e+202 < z < 1.3422271353685726e-58

    1. Initial program 3.8

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

    3. Applied associate-/l*3.9

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

    4. Simplified3.6

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

    6. Applied div-inv3.6

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

    7. Simplified3.5

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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