Average Error: 7.8 → 1.1
Time: 6.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\ \;\;\;\;{\left(\frac{1}{2} \cdot \frac{y \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{x \cdot z}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\
\;\;\;\;{\left(\frac{1}{2} \cdot \frac{y \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{x \cdot z}\right)}^{1}\\

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

\end{array}
double f(double x, double y, double z) {
        double r576217 = x;
        double r576218 = cosh(r576217);
        double r576219 = y;
        double r576220 = r576219 / r576217;
        double r576221 = r576218 * r576220;
        double r576222 = z;
        double r576223 = r576221 / r576222;
        return r576223;
}
↓
double f(double x, double y, double z) {
        double r576224 = y;
        double r576225 = -1.879887519829394e-85;
        bool r576226 = r576224 <= r576225;
        double r576227 = 2.9332543178834714e-108;
        bool r576228 = r576224 <= r576227;
        double r576229 = !r576228;
        bool r576230 = r576226 || r576229;
        double r576231 = 0.5;
        double r576232 = -1.0;
        double r576233 = x;
        double r576234 = r576232 * r576233;
        double r576235 = exp(r576234);
        double r576236 = exp(r576233);
        double r576237 = r576235 + r576236;
        double r576238 = r576224 * r576237;
        double r576239 = z;
        double r576240 = r576233 * r576239;
        double r576241 = r576238 / r576240;
        double r576242 = r576231 * r576241;
        double r576243 = 1.0;
        double r576244 = pow(r576242, r576243);
        double r576245 = r576231 * r576237;
        double r576246 = r576233 / r576224;
        double r576247 = r576245 / r576246;
        double r576248 = r576247 / r576239;
        double r576249 = r576230 ? r576244 : r576248;
        return r576249;
}

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
Target0.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.879887519829394e-85 or 2.9332543178834714e-108 < y

    1. Initial program 14.4

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

    2. Taylor expanded around inf 14.4

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

    3. Simplified14.5

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

    5. Applied *-un-lft-identity14.5

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

    6. Applied div-inv14.5

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

    7. Applied times-frac14.5

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

    8. Applied times-frac1.9

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

    9. Simplified1.9

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

    10. Simplified2.2

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

    12. Applied pow12.2

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

    13. Applied pow12.2

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

    14. Applied pow-prod-down2.2

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

    15. Simplified1.8

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

    if -1.879887519829394e-85 < y < 2.9332543178834714e-108

    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

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\ \;\;\;\;{\left(\frac{1}{2} \cdot \frac{y \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{x \cdot z}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))