Average Error: 7.8 → 0.3
Time: 4.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r420292 = x;
        double r420293 = cosh(r420292);
        double r420294 = y;
        double r420295 = r420294 / r420292;
        double r420296 = r420293 * r420295;
        double r420297 = z;
        double r420298 = r420296 / r420297;
        return r420298;
}
↓
double f(double x, double y, double z) {
        double r420299 = z;
        double r420300 = -3.383191592887116e-08;
        bool r420301 = r420299 <= r420300;
        double r420302 = 3.033555916031553e-25;
        bool r420303 = r420299 <= r420302;
        double r420304 = !r420303;
        bool r420305 = r420301 || r420304;
        double r420306 = x;
        double r420307 = cosh(r420306);
        double r420308 = y;
        double r420309 = r420307 * r420308;
        double r420310 = r420306 * r420299;
        double r420311 = r420309 / r420310;
        double r420312 = r420309 / r420299;
        double r420313 = r420312 / r420306;
        double r420314 = r420305 ? r420311 : r420313;
        return r420314;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.383191592887116e-08 or 3.033555916031553e-25 < z

    1. Initial program 11.4

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

    3. Applied div-inv11.5

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

    5. Applied associate-*r/11.5

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

    6. Applied associate-*l/10.7

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

    7. Simplified10.6

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

    9. Applied div-inv10.7

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

    10. Applied associate-/l*0.4

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

    11. Simplified0.3

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

    if -3.383191592887116e-08 < z < 3.033555916031553e-25

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    5. Applied associate-*r/0.4

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

    6. Applied associate-*l/0.4

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

    7. Simplified0.3

      \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{z}}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.383191592887115717553319195823324783134 \cdot 10^{-8} \lor \neg \left(z \le 3.03355591603155297168608427108120014315 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))