Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.3 → 0.3

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -1.569646754895685203707472211986928556371 \cdot 10^{-6}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.43550247534837939212643563586416747364 \cdot 10^{308}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r560811 = x;
        double r560812 = cosh(r560811);
        double r560813 = y;
        double r560814 = r560813 / r560811;
        double r560815 = r560812 * r560814;
        double r560816 = z;
        double r560817 = r560815 / r560816;
        return r560817;
}

↓

double f(double x, double y, double z) {
        double r560818 = x;
        double r560819 = cosh(r560818);
        double r560820 = y;
        double r560821 = r560820 / r560818;
        double r560822 = r560819 * r560821;
        double r560823 = z;
        double r560824 = r560822 / r560823;
        double r560825 = -1.5696467548956852e-06;
        bool r560826 = r560824 <= r560825;
        double r560827 = r560820 / r560823;
        double r560828 = r560827 / r560818;
        double r560829 = r560819 * r560828;
        double r560830 = 1.4355024753483794e+308;
        bool r560831 = r560824 <= r560830;
        double r560832 = 1.0;
        double r560833 = r560832 / r560823;
        double r560834 = r560833 / r560818;
        double r560835 = r560820 * r560834;
        double r560836 = r560819 * r560835;
        double r560837 = r560831 ? r560824 : r560836;
        double r560838 = r560826 ? r560829 : r560837;
        return r560838;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.3
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/ (* (cosh x) (/ y x)) z) < -1.5696467548956852e-06

   1. Initial program 12.2

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

   3. Applied *-un-lft-identity 12.2

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

   4. Applied times-frac 12.2

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

   5. Simplified 12.2

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

   6. Simplified 11.5

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

   8. Applied div-inv 12.6

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

   10. Applied add-cube-cbrt 12.6

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

   11. Applied times-frac 12.7

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

   12. Simplified 12.7

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

   13. Simplified 12.7

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

   15. Applied associate-*l/ 12.6

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

   16. Applied associate-*r/ 0.4

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

   17. Simplified 0.3

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

   if -1.5696467548956852e-06 < (/ (* (cosh x) (/ y x)) z) < 1.4355024753483794e+308

   1. Initial program 0.2

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

   if 1.4355024753483794e+308 < (/ (* (cosh x) (/ y x)) z)

   1. Initial program 63.9

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

   3. Applied *-un-lft-identity 63.9

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

   4. Applied times-frac 63.7

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

   5. Simplified 63.7

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

   6. Simplified 0.4

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

   8. Applied div-inv 0.5

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

   10. Applied add-cube-cbrt 0.5

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

   11. Applied times-frac 0.6

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

   12. Simplified 0.6

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

   13. Simplified 0.6

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

   15. Applied associate-*l/ 0.5

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

   16. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -1.569646754895685203707472211986928556371 \cdot 10^{-6}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.43550247534837939212643563586416747364 \cdot 10^{308}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \frac{\frac{1}{z}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))