Average Error: 7.9 → 1.0
Time: 3.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.1538486122792733 \cdot 10^{43}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)\right) + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 4.00822008275982678 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.1538486122792733 \cdot 10^{43}:\\
\;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)\right) + \frac{y}{x \cdot z}\\

\mathbf{elif}\;y \le 4.00822008275982678 \cdot 10^{-26}:\\
\;\;\;\;x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right) + \frac{\frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r575201 = x;
        double r575202 = cosh(r575201);
        double r575203 = y;
        double r575204 = r575203 / r575201;
        double r575205 = r575202 * r575204;
        double r575206 = z;
        double r575207 = r575205 / r575206;
        return r575207;
}
↓
double f(double x, double y, double z) {
        double r575208 = y;
        double r575209 = -2.1538486122792733e+43;
        bool r575210 = r575208 <= r575209;
        double r575211 = x;
        double r575212 = cbrt(r575211);
        double r575213 = r575212 * r575212;
        double r575214 = 0.5;
        double r575215 = z;
        double r575216 = r575208 / r575215;
        double r575217 = r575214 * r575216;
        double r575218 = r575212 * r575217;
        double r575219 = r575213 * r575218;
        double r575220 = r575211 * r575215;
        double r575221 = r575208 / r575220;
        double r575222 = r575219 + r575221;
        double r575223 = 4.008220082759827e-26;
        bool r575224 = r575208 <= r575223;
        double r575225 = r575211 * r575217;
        double r575226 = r575208 / r575211;
        double r575227 = r575226 / r575215;
        double r575228 = r575225 + r575227;
        double r575229 = -1.0;
        double r575230 = r575229 * r575211;
        double r575231 = exp(r575230);
        double r575232 = exp(r575211);
        double r575233 = r575231 + r575232;
        double r575234 = r575214 * r575233;
        double r575235 = r575215 * r575211;
        double r575236 = r575235 / r575208;
        double r575237 = r575234 / r575236;
        double r575238 = r575224 ? r575228 : r575237;
        double r575239 = r575210 ? r575222 : r575238;
        return r575239;
}

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

Original7.9
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.1538486122792733e+43

    1. Initial program 25.0

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

    2. Taylor expanded around 0 1.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity1.6

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

    5. Applied times-frac1.6

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

    6. Applied associate-*r*1.6

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

    7. Simplified1.6

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

    9. Applied associate-*l*1.6

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

    11. Applied add-cube-cbrt1.6

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

    12. Applied associate-*l*1.6

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

    if -2.1538486122792733e+43 < y < 4.008220082759827e-26

    1. Initial program 0.5

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

    2. Taylor expanded around 0 11.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity11.1

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

    5. Applied times-frac11.1

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

    6. Applied associate-*r*11.1

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

    7. Simplified11.1

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

    9. Applied associate-*l*11.1

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

    11. Applied associate-/r*1.1

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

    if 4.008220082759827e-26 < y

    1. Initial program 19.7

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.1538486122792733 \cdot 10^{43}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right)\right) + \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 4.00822008275982678 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} \cdot \frac{y}{z}\right) + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))