Average Error: 7.8 → 0.6
Time: 17.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.750320107203263883619067878118478993193 \cdot 10^{90} \lor \neg \left(z \le 18485054819100778299392\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.750320107203263883619067878118478993193 \cdot 10^{90} \lor \neg \left(z \le 18485054819100778299392\right):\\
\;\;\;\;\frac{y}{\frac{z \cdot x}{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r312081 = x;
        double r312082 = cosh(r312081);
        double r312083 = y;
        double r312084 = r312083 / r312081;
        double r312085 = r312082 * r312084;
        double r312086 = z;
        double r312087 = r312085 / r312086;
        return r312087;
}
↓
double f(double x, double y, double z) {
        double r312088 = z;
        double r312089 = -9.750320107203264e+90;
        bool r312090 = r312088 <= r312089;
        double r312091 = 1.8485054819100778e+22;
        bool r312092 = r312088 <= r312091;
        double r312093 = !r312092;
        bool r312094 = r312090 || r312093;
        double r312095 = y;
        double r312096 = x;
        double r312097 = r312088 * r312096;
        double r312098 = 0.5;
        double r312099 = -r312096;
        double r312100 = exp(r312099);
        double r312101 = exp(r312096);
        double r312102 = r312100 + r312101;
        double r312103 = r312098 * r312102;
        double r312104 = r312097 / r312103;
        double r312105 = r312095 / r312104;
        double r312106 = r312101 + r312100;
        double r312107 = r312106 * r312095;
        double r312108 = r312107 / r312088;
        double r312109 = 2.0;
        double r312110 = r312109 * r312096;
        double r312111 = r312108 / r312110;
        double r312112 = r312094 ? r312105 : r312111;
        return r312112;
}

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

Original7.8
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -9.750320107203264e+90 or 1.8485054819100778e+22 < z

    1. Initial program 13.2

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

    3. Applied cosh-def13.2

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

    4. Applied frac-times13.2

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

    5. Applied associate-/l/0.4

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

    7. Applied add-sqr-sqrt1.2

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

    8. Applied associate-*l*0.9

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

    9. Taylor expanded around inf 0.4

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

    10. Simplified0.3

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

    if -9.750320107203264e+90 < z < 1.8485054819100778e+22

    1. Initial program 1.3

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

    3. Applied cosh-def1.3

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

    4. Applied frac-times1.3

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

    5. Applied associate-/l/14.4

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

    7. Applied associate-/r*1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.750320107203263883619067878118478993193 \cdot 10^{90} \lor \neg \left(z \le 18485054819100778299392\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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))