Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.3 → 0.6

Time: 13.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.615131604836802622424408328927110030211 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{elif}\;z \le 3.366351636300589121970729979201097827583 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{\frac{y}{x}}{z}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r574934 = x;
        double r574935 = cosh(r574934);
        double r574936 = y;
        double r574937 = r574936 / r574934;
        double r574938 = r574935 * r574937;
        double r574939 = z;
        double r574940 = r574938 / r574939;
        return r574940;
}

↓

double f(double x, double y, double z) {
        double r574941 = z;
        double r574942 = -1.6151316048368026e-74;
        bool r574943 = r574941 <= r574942;
        double r574944 = x;
        double r574945 = exp(r574944);
        double r574946 = -r574944;
        double r574947 = exp(r574946);
        double r574948 = r574945 + r574947;
        double r574949 = y;
        double r574950 = r574948 * r574949;
        double r574951 = 2.0;
        double r574952 = r574951 * r574944;
        double r574953 = r574941 * r574952;
        double r574954 = r574950 / r574953;
        double r574955 = 3.366351636300589e-14;
        bool r574956 = r574941 <= r574955;
        double r574957 = r574949 / r574944;
        double r574958 = r574957 / r574941;
        double r574959 = r574948 * r574958;
        double r574960 = r574959 / r574951;
        double r574961 = cosh(r574944);
        double r574962 = r574961 * r574949;
        double r574963 = 1.0;
        double r574964 = r574963 / r574944;
        double r574965 = r574964 / r574941;
        double r574966 = r574962 * r574965;
        double r574967 = r574956 ? r574960 : r574966;
        double r574968 = r574943 ? r574954 : r574967;
        return r574968;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	8.3
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if z < -1.6151316048368026e-74

   1. Initial program 11.7

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

   3. Applied cosh-def 11.7

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

   4. Applied frac-times 11.7

      \[\leadsto \frac{\color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2 \cdot x}}}{z}\]

   5. Applied associate-/l/ 0.8

      \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}}\]

   if -1.6151316048368026e-74 < z < 3.366351636300589e-14

   1. Initial program 0.3

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

   3. Applied *-un-lft-identity 0.3

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

   4. Applied times-frac 0.3

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

   5. Simplified 0.3

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

   7. Applied cosh-def 0.3

      \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{\frac{y}{x}}{z}\]

   8. Applied associate-*l/ 0.4

      \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{\frac{y}{x}}{z}}{2}}\]

   if 3.366351636300589e-14 < z

   1. Initial program 11.0

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

   3. Applied *-un-lft-identity 11.0

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

   4. Applied times-frac 11.0

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

   5. Simplified 11.0

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

   7. Applied *-un-lft-identity 11.0

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

   8. Applied div-inv 11.1

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

   9. Applied times-frac 0.5

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

   10. Applied associate-*r* 0.5

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

   11. Simplified 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.615131604836802622424408328927110030211 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{elif}\;z \le 3.366351636300589121970729979201097827583 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{\frac{y}{x}}{z}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))