Average Error: 7.7 → 0.8
Time: 20.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.66967556320118367307962221628093140978 \cdot 10^{106} \lor \neg \left(z \le 1.580946681139865959507055911418667078904 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(e^{-x} + e^{x}\right) \cdot y}{x}}{2}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.66967556320118367307962221628093140978 \cdot 10^{106} \lor \neg \left(z \le 1.580946681139865959507055911418667078904 \cdot 10^{-36}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r302184 = x;
        double r302185 = cosh(r302184);
        double r302186 = y;
        double r302187 = r302186 / r302184;
        double r302188 = r302185 * r302187;
        double r302189 = z;
        double r302190 = r302188 / r302189;
        return r302190;
}
↓
double f(double x, double y, double z) {
        double r302191 = z;
        double r302192 = -1.6696755632011837e+106;
        bool r302193 = r302191 <= r302192;
        double r302194 = 1.580946681139866e-36;
        bool r302195 = r302191 <= r302194;
        double r302196 = !r302195;
        bool r302197 = r302193 || r302196;
        double r302198 = x;
        double r302199 = cosh(r302198);
        double r302200 = y;
        double r302201 = r302199 * r302200;
        double r302202 = r302191 * r302198;
        double r302203 = r302201 / r302202;
        double r302204 = -r302198;
        double r302205 = exp(r302204);
        double r302206 = exp(r302198);
        double r302207 = r302205 + r302206;
        double r302208 = r302207 * r302200;
        double r302209 = r302208 / r302198;
        double r302210 = 2.0;
        double r302211 = r302209 / r302210;
        double r302212 = r302211 / r302191;
        double r302213 = r302197 ? r302203 : r302212;
        return r302213;
}

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

Original7.7
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.6696755632011837e+106 or 1.580946681139866e-36 < z

    1. Initial program 12.3

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

    3. Applied associate-*r/12.3

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

    4. Applied associate-/l/0.3

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

    if -1.6696755632011837e+106 < z < 1.580946681139866e-36

    1. Initial program 1.5

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

    3. Applied div-inv1.6

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

    4. Applied associate-*r*1.6

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

    6. Applied cosh-def1.6

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

    7. Applied associate-*l/1.6

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

    8. Applied associate-*l/1.6

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

    9. Simplified1.5

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.66967556320118367307962221628093140978 \cdot 10^{106} \lor \neg \left(z \le 1.580946681139865959507055911418667078904 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(e^{-x} + e^{x}\right) \cdot y}{x}}{2}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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