Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.6

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3530667918855812 \cdot 10^{-68} \lor \neg \left(x \le 1.46451474323560222 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{\frac{\sin y}{y}}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r451331 = x;
        double r451332 = y;
        double r451333 = sin(r451332);
        double r451334 = r451333 / r451332;
        double r451335 = r451331 * r451334;
        double r451336 = z;
        double r451337 = r451335 / r451336;
        return r451337;
}

↓

double f(double x, double y, double z) {
        double r451338 = x;
        double r451339 = -1.3530667918855812e-68;
        bool r451340 = r451338 <= r451339;
        double r451341 = 1.4645147432356022e-46;
        bool r451342 = r451338 <= r451341;
        double r451343 = !r451342;
        bool r451344 = r451340 || r451343;
        double r451345 = y;
        double r451346 = sin(r451345);
        double r451347 = r451345 / r451346;
        double r451348 = r451338 / r451347;
        double r451349 = z;
        double r451350 = r451348 / r451349;
        double r451351 = 1.0;
        double r451352 = r451349 / r451338;
        double r451353 = r451346 / r451345;
        double r451354 = r451352 / r451353;
        double r451355 = r451351 / r451354;
        double r451356 = r451344 ? r451350 : r451355;
        return r451356;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3530667918855812e-68 or 1.4645147432356022e-46 < x

   1. Initial program 0.4

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

   3. Applied clear-num 0.4

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

   5. Applied *-un-lft-identity 0.4

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

   6. Applied associate-/r* 0.4

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

   7. Simplified 0.4

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

   if -1.3530667918855812e-68 < x < 1.4645147432356022e-46

   1. Initial program 5.7

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

   3. Applied clear-num 5.7

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

   5. Applied clear-num 6.3

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

   6. Simplified 0.8

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{x}}{\frac{\sin y}{y}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3530667918855812 \cdot 10^{-68} \lor \neg \left(x \le 1.46451474323560222 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))