Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.6

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.38157623106928596 \cdot 10^{54}:\\ \;\;\;\;\cosh x \cdot \left(\frac{\sqrt{1}}{x} \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;y \le 3.006240253092873 \cdot 10^{32}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(\frac{\frac{1}{x}}{z} \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r553332 = x;
        double r553333 = cosh(r553332);
        double r553334 = y;
        double r553335 = r553334 / r553332;
        double r553336 = r553333 * r553335;
        double r553337 = z;
        double r553338 = r553336 / r553337;
        return r553338;
}

↓

double f(double x, double y, double z) {
        double r553339 = y;
        double r553340 = -3.381576231069286e+54;
        bool r553341 = r553339 <= r553340;
        double r553342 = x;
        double r553343 = cosh(r553342);
        double r553344 = 1.0;
        double r553345 = sqrt(r553344);
        double r553346 = r553345 / r553342;
        double r553347 = z;
        double r553348 = r553339 / r553347;
        double r553349 = r553346 * r553348;
        double r553350 = r553343 * r553349;
        double r553351 = 3.006240253092873e+32;
        bool r553352 = r553339 <= r553351;
        double r553353 = r553339 / r553342;
        double r553354 = r553353 / r553347;
        double r553355 = r553343 * r553354;
        double r553356 = r553344 / r553342;
        double r553357 = r553356 / r553347;
        double r553358 = r553357 * r553339;
        double r553359 = r553343 * r553358;
        double r553360 = r553352 ? r553355 : r553359;
        double r553361 = r553341 ? r553350 : r553360;
        return r553361;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	0.4
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if y < -3.381576231069286e+54

   1. Initial program 27.1

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

   3. Applied *-un-lft-identity 27.1

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

   4. Applied times-frac 27.0

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

   5. Simplified 27.0

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

   6. Simplified 0.3

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

   8. Applied clear-num 0.4

      \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}}\]
   9. Using strategy rm

   10. Applied div-inv 0.5

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

   11. Applied add-cube-cbrt 0.5

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

   12. Applied times-frac 0.4

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

   13. Simplified 0.4

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

   14. Simplified 0.3

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

   16. Applied add-sqr-sqrt 0.3

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

   17. Applied times-frac 0.4

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

   18. Applied associate-*l* 0.4

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

   19. Simplified 0.4

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

   if -3.381576231069286e+54 < y < 3.006240253092873e+32

   1. Initial program 0.6

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

   3. Applied *-un-lft-identity 0.6

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

   4. Applied times-frac 0.7

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

   5. Simplified 0.7

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

   6. Simplified 9.9

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

   8. Applied associate-/r* 0.7

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

   if 3.006240253092873e+32 < y

   1. Initial program 24.9

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

   3. Applied *-un-lft-identity 24.9

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

   4. Applied times-frac 24.8

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

   5. Simplified 24.8

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

   6. Simplified 0.3

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

   8. Applied clear-num 0.4

      \[\leadsto \cosh x \cdot \color{blue}{\frac{1}{\frac{x \cdot z}{y}}}\]
   9. Using strategy rm

   10. Applied div-inv 0.5

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

   11. Applied add-cube-cbrt 0.5

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

   12. Applied times-frac 0.5

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

   13. Simplified 0.5

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

   14. Simplified 0.4

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

   16. Applied associate-/r* 0.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.38157623106928596 \cdot 10^{54}:\\ \;\;\;\;\cosh x \cdot \left(\frac{\sqrt{1}}{x} \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;y \le 3.006240253092873 \cdot 10^{32}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(\frac{\frac{1}{x}}{z} \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))