Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.9 → 0.2

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.579176583523817341408656991031950031801 \cdot 10^{-58} \lor \neg \left(z \le 14047118970.9252910614013671875\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r557964 = x;
        double r557965 = y;
        double r557966 = sin(r557965);
        double r557967 = r557966 / r557965;
        double r557968 = r557964 * r557967;
        double r557969 = z;
        double r557970 = r557968 / r557969;
        return r557970;
}

↓

double f(double x, double y, double z) {
        double r557971 = z;
        double r557972 = -1.5791765835238173e-58;
        bool r557973 = r557971 <= r557972;
        double r557974 = 14047118970.925291;
        bool r557975 = r557971 <= r557974;
        double r557976 = !r557975;
        bool r557977 = r557973 || r557976;
        double r557978 = x;
        double r557979 = 1.0;
        double r557980 = y;
        double r557981 = r557979 / r557980;
        double r557982 = sin(r557980);
        double r557983 = r557979 / r557982;
        double r557984 = r557981 / r557983;
        double r557985 = r557978 * r557984;
        double r557986 = r557985 / r557971;
        double r557987 = r557982 / r557980;
        double r557988 = r557971 / r557987;
        double r557989 = r557978 / r557988;
        double r557990 = r557977 ? r557986 : r557989;
        return r557990;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.9
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.5791765835238173e-58 or 14047118970.925291 < z

   1. Initial program 0.2

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

   3. Applied clear-num 0.2

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

   5. Applied div-inv 0.3

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

   6. Applied associate-/r* 0.2

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

   if -1.5791765835238173e-58 < z < 14047118970.925291

   1. Initial program 6.6

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

   3. Applied associate-/l* 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.579176583523817341408656991031950031801 \cdot 10^{-58} \lor \neg \left(z \le 14047118970.9252910614013671875\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))