Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 3.0 → 0.8

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.1728892342568057 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{y} \cdot \sin y\right)}{z}\\ \mathbf{elif}\;z \le 1.32101974044890363 \cdot 10^{187}:\\ \;\;\;\;\frac{1 \cdot x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot \frac{x}{\frac{y}{\sin y}}\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r522400 = x;
        double r522401 = y;
        double r522402 = sin(r522401);
        double r522403 = r522402 / r522401;
        double r522404 = r522400 * r522403;
        double r522405 = z;
        double r522406 = r522404 / r522405;
        return r522406;
}

↓

double f(double x, double y, double z) {
        double r522407 = z;
        double r522408 = -1.1728892342568057e-27;
        bool r522409 = r522407 <= r522408;
        double r522410 = x;
        double r522411 = 1.0;
        double r522412 = y;
        double r522413 = r522411 / r522412;
        double r522414 = sin(r522412);
        double r522415 = r522413 * r522414;
        double r522416 = r522410 * r522415;
        double r522417 = r522416 / r522407;
        double r522418 = 1.3210197404489036e+187;
        bool r522419 = r522407 <= r522418;
        double r522420 = r522411 * r522410;
        double r522421 = r522412 / r522414;
        double r522422 = r522407 * r522421;
        double r522423 = r522420 / r522422;
        double r522424 = r522410 / r522421;
        double r522425 = r522411 * r522424;
        double r522426 = r522411 / r522407;
        double r522427 = r522425 * r522426;
        double r522428 = r522419 ? r522423 : r522427;
        double r522429 = r522409 ? r522417 : r522428;
        return r522429;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.0
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1.1728892342568057e-27

   1. Initial program 0.2

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

   3. Applied clear-num 0.3

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

   5. Applied div-inv 0.4

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

   6. Applied add-cube-cbrt 0.4

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

   7. Applied times-frac 0.3

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

   8. Simplified 0.3

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

   9. Simplified 0.3

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

   if -1.1728892342568057e-27 < z < 1.3210197404489036e+187

   1. Initial program 4.8

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

   3. Applied clear-num 4.9

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

   5. Applied *-un-lft-identity 4.9

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

   6. Applied associate-*l* 4.9

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

   7. Simplified 4.8

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

   9. Applied associate-*r/ 4.8

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

   10. Applied associate-/l/ 1.1

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

   if 1.3210197404489036e+187 < z

   1. Initial program 0.1

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

   3. Applied clear-num 0.1

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

   5. Applied *-un-lft-identity 0.1

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

   6. Applied associate-*l* 0.1

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

   7. Simplified 0.1

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

   9. Applied div-inv 0.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1728892342568057 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{y} \cdot \sin y\right)}{z}\\ \mathbf{elif}\;z \le 1.32101974044890363 \cdot 10^{187}:\\ \;\;\;\;\frac{1 \cdot x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot \frac{x}{\frac{y}{\sin y}}\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))