Average Error: 7.4 → 0.7
Time: 16.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.936045909707795 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y\right) \cdot \frac{\frac{1}{z}}{x + x}\\ \mathbf{elif}\;z \le 7.694949687069597 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{x \cdot z} \cdot \frac{1}{2}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.936045909707795 \cdot 10^{+96}:\\
\;\;\;\;\left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y\right) \cdot \frac{\frac{1}{z}}{x + x}\\

\mathbf{elif}\;z \le 7.694949687069597 \cdot 10^{-19}:\\
\;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right) \cdot \frac{y}{z}}{x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r28657119 = x;
        double r28657120 = cosh(r28657119);
        double r28657121 = y;
        double r28657122 = r28657121 / r28657119;
        double r28657123 = r28657120 * r28657122;
        double r28657124 = z;
        double r28657125 = r28657123 / r28657124;
        return r28657125;
}
↓
double f(double x, double y, double z) {
        double r28657126 = z;
        double r28657127 = -5.936045909707795e+96;
        bool r28657128 = r28657126 <= r28657127;
        double r28657129 = x;
        double r28657130 = exp(r28657129);
        double r28657131 = 1.0;
        double r28657132 = r28657131 / r28657130;
        double r28657133 = r28657130 + r28657132;
        double r28657134 = y;
        double r28657135 = r28657133 * r28657134;
        double r28657136 = r28657131 / r28657126;
        double r28657137 = r28657129 + r28657129;
        double r28657138 = r28657136 / r28657137;
        double r28657139 = r28657135 * r28657138;
        double r28657140 = 7.694949687069597e-19;
        bool r28657141 = r28657126 <= r28657140;
        double r28657142 = 0.5;
        double r28657143 = r28657142 / r28657130;
        double r28657144 = r28657130 * r28657142;
        double r28657145 = r28657143 + r28657144;
        double r28657146 = r28657134 / r28657126;
        double r28657147 = r28657145 * r28657146;
        double r28657148 = r28657147 / r28657129;
        double r28657149 = r28657129 * r28657126;
        double r28657150 = r28657135 / r28657149;
        double r28657151 = r28657150 * r28657142;
        double r28657152 = r28657141 ? r28657148 : r28657151;
        double r28657153 = r28657128 ? r28657139 : r28657152;
        return r28657153;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.936045909707795e+96

    1. Initial program 13.7

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

    3. Applied cosh-def13.7

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

    4. Applied frac-times13.7

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

    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 clear-num0.8

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

    9. Applied div-inv0.8

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

    10. Applied add-cube-cbrt0.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(z \cdot \left(2 \cdot x\right)\right) \cdot \frac{1}{\left(e^{x} + e^{-x}\right) \cdot y}}\]

    11. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{z \cdot \left(2 \cdot x\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(e^{x} + e^{-x}\right) \cdot y}}}\]

    12. Simplified0.5

      \[\leadsto \color{blue}{\frac{\frac{1}{z}}{x + x}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(e^{x} + e^{-x}\right) \cdot y}}\]

    13. Simplified0.5

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

    if -5.936045909707795e+96 < z < 7.694949687069597e-19

    1. Initial program 1.5

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

    2. Taylor expanded around inf 15.9

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

    3. Simplified1.2

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

    if 7.694949687069597e-19 < z

    1. Initial program 10.6

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

    3. Applied cosh-def10.6

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

    4. Applied frac-times10.6

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

    5. Applied associate-/l/0.3

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

    7. Applied clear-num0.7

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

    9. Applied times-frac10.0

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

    10. Applied associate-/r*9.7

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

    11. Simplified9.7

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

    12. Taylor expanded around inf 0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.936045909707795 \cdot 10^{+96}:\\ \;\;\;\;\left(\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y\right) \cdot \frac{\frac{1}{z}}{x + x}\\ \mathbf{elif}\;z \le 7.694949687069597 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right) \cdot \frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot y}{x \cdot z} \cdot \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :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))