Average Error: 7.4 → 0.6
Time: 3.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.2558391828450102 \cdot 10^{-46} \lor \neg \left(z \le 2.98505390964268964 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{-1 \cdot x}, \frac{1}{2} \cdot e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.2558391828450102 \cdot 10^{-46} \lor \neg \left(z \le 2.98505390964268964 \cdot 10^{-67}\right):\\
\;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \left(x \cdot 2\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r618611 = x;
        double r618612 = cosh(r618611);
        double r618613 = y;
        double r618614 = r618613 / r618611;
        double r618615 = r618612 * r618614;
        double r618616 = z;
        double r618617 = r618615 / r618616;
        return r618617;
}
↓
double f(double x, double y, double z) {
        double r618618 = z;
        double r618619 = -4.25583918284501e-46;
        bool r618620 = r618618 <= r618619;
        double r618621 = 2.9850539096426896e-67;
        bool r618622 = r618618 <= r618621;
        double r618623 = !r618622;
        bool r618624 = r618620 || r618623;
        double r618625 = y;
        double r618626 = x;
        double r618627 = exp(r618626);
        double r618628 = -r618626;
        double r618629 = exp(r618628);
        double r618630 = r618627 + r618629;
        double r618631 = r618625 * r618630;
        double r618632 = 2.0;
        double r618633 = r618626 * r618632;
        double r618634 = r618618 * r618633;
        double r618635 = r618631 / r618634;
        double r618636 = 0.5;
        double r618637 = -1.0;
        double r618638 = r618637 * r618626;
        double r618639 = exp(r618638);
        double r618640 = r618636 * r618627;
        double r618641 = fma(r618636, r618639, r618640);
        double r618642 = r618626 / r618625;
        double r618643 = r618641 / r618642;
        double r618644 = r618643 / r618618;
        double r618645 = r618624 ? r618635 : r618644;
        return r618645;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.4
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.25583918284501e-46 or 2.9850539096426896e-67 < z

    1. Initial program 10.0

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

    3. Applied *-commutative10.0

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

    5. Applied cosh-def10.0

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

    6. Applied frac-times10.0

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

    7. Applied associate-/l/0.7

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

    if -4.25583918284501e-46 < z < 2.9850539096426896e-67

    1. Initial program 0.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.2558391828450102 \cdot 10^{-46} \lor \neg \left(z \le 2.98505390964268964 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \left(x \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{-1 \cdot x}, \frac{1}{2} \cdot e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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