Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 1.0

Time: 15.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -7.807003819640308 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 1.0579617958619699 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\cosh x}{z}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r407828 = x;
        double r407829 = cosh(r407828);
        double r407830 = y;
        double r407831 = r407830 / r407828;
        double r407832 = r407829 * r407831;
        double r407833 = z;
        double r407834 = r407832 / r407833;
        return r407834;
}

↓

double f(double x, double y, double z) {
        double r407835 = y;
        double r407836 = -7.807003819640308e-86;
        bool r407837 = r407835 <= r407836;
        double r407838 = x;
        double r407839 = cosh(r407838);
        double r407840 = r407839 * r407835;
        double r407841 = z;
        double r407842 = r407840 / r407841;
        double r407843 = r407842 / r407838;
        double r407844 = 1.0579617958619699e-66;
        bool r407845 = r407835 <= r407844;
        double r407846 = r407839 / r407841;
        double r407847 = r407838 / r407835;
        double r407848 = r407846 / r407847;
        double r407849 = 0.5;
        double r407850 = r407838 * r407835;
        double r407851 = r407850 / r407841;
        double r407852 = r407849 * r407851;
        double r407853 = r407838 * r407841;
        double r407854 = r407835 / r407853;
        double r407855 = r407852 + r407854;
        double r407856 = r407845 ? r407848 : r407855;
        double r407857 = r407837 ? r407843 : r407856;
        return r407857;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	0.4
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.03853053593515302 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -7.807003819640308e-86

   1. Initial program 14.5

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

   3. Applied associate-*r/ 14.5

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

   4. Applied associate-/l/ 1.4

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

   6. Applied associate-/r* 1.7

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

   if -7.807003819640308e-86 < y < 1.0579617958619699e-66

   1. Initial program 0.3

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

   3. Applied associate-*r/ 0.3

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

   4. Applied associate-/l/ 12.7

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

   6. Applied clear-num 13.2

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

   8. Applied *-un-lft-identity 13.2

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

   9. Applied add-cube-cbrt 13.2

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

   10. Applied times-frac 13.2

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

   11. Simplified 13.2

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

   12. Simplified 0.4

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

   if 1.0579617958619699e-66 < y

   1. Initial program 15.8

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

   3. Applied associate-*r/ 15.8

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

   4. Applied associate-/l/ 1.1

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

   6. Applied clear-num 1.2

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

   8. Applied *-un-lft-identity 1.2

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

   9. Applied add-cube-cbrt 1.2

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

   10. Applied times-frac 1.2

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

   11. Simplified 1.2

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

   12. Simplified 14.4

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

   13. Taylor expanded around 0 1.7

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.807003819640308 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 1.0579617958619699 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\cosh x}{z}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))