Average Error: 7.8 → 6.8
Time: 11.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\cosh x \cdot \frac{y}{x \cdot z}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\cosh x \cdot \frac{y}{x \cdot z}
double f(double x, double y, double z) {
        double r364770 = x;
        double r364771 = cosh(r364770);
        double r364772 = y;
        double r364773 = r364772 / r364770;
        double r364774 = r364771 * r364773;
        double r364775 = z;
        double r364776 = r364774 / r364775;
        return r364776;
}
↓
double f(double x, double y, double z) {
        double r364777 = x;
        double r364778 = cosh(r364777);
        double r364779 = y;
        double r364780 = z;
        double r364781 = r364777 * r364780;
        double r364782 = r364779 / r364781;
        double r364783 = r364778 * r364782;
        return r364783;
}

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
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -12057758525001.123 or 42042.628727599236 < z

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac12.2

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

    5. Simplified12.2

      \[\leadsto \color{blue}{\cosh x} \cdot \frac{\frac{y}{x}}{z}\]

    6. Simplified0.3

      \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}}\]

    if -12057758525001.123 < z < 42042.628727599236

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

      \[\leadsto \color{blue}{\cosh x} \cdot \frac{\frac{y}{x}}{z}\]

    6. Simplified17.9

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

    8. Applied *-un-lft-identity17.9

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

    9. Applied times-frac0.4

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

    11. Applied pow10.4

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

    12. Applied pow10.4

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

    13. Applied pow-prod-down0.4

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

    14. Simplified0.3

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

  4. Final simplification6.8

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

Reproduce

herbie shell --seed 2019303 
(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))