Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.9 → 0.2

Time: 6.7s

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 r551029 = x;
        double r551030 = y;
        double r551031 = sin(r551030);
        double r551032 = r551031 / r551030;
        double r551033 = r551029 * r551032;
        double r551034 = z;
        double r551035 = r551033 / r551034;
        return r551035;
}

↓

double f(double x, double y, double z) {
        double r551036 = z;
        double r551037 = -1.5791765835238173e-58;
        bool r551038 = r551036 <= r551037;
        double r551039 = 14047118970.925291;
        bool r551040 = r551036 <= r551039;
        double r551041 = !r551040;
        bool r551042 = r551038 || r551041;
        double r551043 = x;
        double r551044 = 1.0;
        double r551045 = y;
        double r551046 = r551044 / r551045;
        double r551047 = sin(r551045);
        double r551048 = r551044 / r551047;
        double r551049 = r551046 / r551048;
        double r551050 = r551043 * r551049;
        double r551051 = r551050 / r551036;
        double r551052 = r551047 / r551045;
        double r551053 = r551036 / r551052;
        double r551054 = r551043 / r551053;
        double r551055 = r551042 ? r551051 : r551054;
        return r551055;
}

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 +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))