Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.4 → 0.5

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.1569808277222708 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}} \cdot \sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}\\ \mathbf{elif}\;z \le 2.0130873820056925 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}} \cdot \sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r21968373 = x;
        double r21968374 = cosh(r21968373);
        double r21968375 = y;
        double r21968376 = r21968375 / r21968373;
        double r21968377 = r21968374 * r21968376;
        double r21968378 = z;
        double r21968379 = r21968377 / r21968378;
        return r21968379;
}

↓

double f(double x, double y, double z) {
        double r21968380 = z;
        double r21968381 = -3.1569808277222708e-65;
        bool r21968382 = r21968380 <= r21968381;
        double r21968383 = y;
        double r21968384 = x;
        double r21968385 = r21968384 * r21968380;
        double r21968386 = exp(r21968384);
        double r21968387 = 0.5;
        double r21968388 = r21968387 / r21968386;
        double r21968389 = fma(r21968386, r21968387, r21968388);
        double r21968390 = sqrt(r21968389);
        double r21968391 = r21968385 / r21968390;
        double r21968392 = r21968383 / r21968391;
        double r21968393 = r21968392 * r21968390;
        double r21968394 = 2.0130873820056925e-20;
        bool r21968395 = r21968380 <= r21968394;
        double r21968396 = fma(r21968387, r21968386, r21968388);
        double r21968397 = r21968380 / r21968396;
        double r21968398 = r21968383 / r21968397;
        double r21968399 = r21968398 / r21968384;
        double r21968400 = r21968395 ? r21968399 : r21968393;
        double r21968401 = r21968382 ? r21968393 : r21968400;
        return r21968401;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -3.1569808277222708e-65 or 2.0130873820056925e-20 < z

   1. Initial program 10.3

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

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right) \cdot y}{x \cdot z}}\]

   3. Simplified 10.3

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{z}}\]

   4. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot \frac{1}{e^{x}}\right) \cdot y}{x \cdot z}}\]

   5. Simplified 10.3

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 10.3

      \[\leadsto \frac{\frac{y}{x}}{\frac{z}{\color{blue}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}}\]

   8. Applied *-un-lft-identity 10.3

      \[\leadsto \frac{\frac{y}{x}}{\frac{\color{blue}{1 \cdot z}}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]

   9. Applied times-frac 10.3

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}} \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}}\]

   10. Applied *-un-lft-identity 10.3

       \[\leadsto \frac{\frac{y}{\color{blue}{1 \cdot x}}}{\frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}} \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]

   11. Applied *-un-lft-identity 10.3

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot y}}{1 \cdot x}}{\frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}} \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]

   12. Applied times-frac 10.3

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{y}{x}}}{\frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}} \cdot \frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]

   13. Applied times-frac 10.3

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}} \cdot \frac{\frac{y}{x}}{\frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}}\]

   14. Simplified 10.3

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}} \cdot \frac{\frac{y}{x}}{\frac{z}{\sqrt{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]

   15. Simplified 0.5

       \[\leadsto \sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)} \cdot \color{blue}{\frac{y}{\frac{z \cdot x}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}}}\]

   if -3.1569808277222708e-65 < z < 2.0130873820056925e-20

   1. Initial program 0.3

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

   2. Taylor expanded around inf 22.4

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right) \cdot y}{x \cdot z}}\]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{z}}\]

   4. Taylor expanded around inf 22.4

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot \frac{1}{e^{x}}\right) \cdot y}{x \cdot z}}\]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]
   6. Using strategy rm

   7. Applied div-inv 0.5

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{1}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}\]
   8. Using strategy rm

   9. Applied associate-*l/ 0.5

      \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{x}}\]

   10. Simplified 0.4

       \[\leadsto \frac{\color{blue}{\frac{y}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.1569808277222708 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}} \cdot \sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}\\ \mathbf{elif}\;z \le 2.0130873820056925 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}} \cdot \sqrt{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))