Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.4 → 1.1

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.230194912459417 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot \left(\sin y \cdot x\right)}{z}\\ \mathbf{elif}\;x \le 5.118933034108623 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot \left(\sin y \cdot x\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r20939020 = x;
        double r20939021 = y;
        double r20939022 = sin(r20939021);
        double r20939023 = r20939022 / r20939021;
        double r20939024 = r20939020 * r20939023;
        double r20939025 = z;
        double r20939026 = r20939024 / r20939025;
        return r20939026;
}

↓

double f(double x, double y, double z) {
        double r20939027 = x;
        double r20939028 = -8.230194912459417e+102;
        bool r20939029 = r20939027 <= r20939028;
        double r20939030 = 1.0;
        double r20939031 = y;
        double r20939032 = r20939030 / r20939031;
        double r20939033 = sin(r20939031);
        double r20939034 = r20939033 * r20939027;
        double r20939035 = r20939032 * r20939034;
        double r20939036 = z;
        double r20939037 = r20939035 / r20939036;
        double r20939038 = 5.118933034108623e-123;
        bool r20939039 = r20939027 <= r20939038;
        double r20939040 = r20939033 / r20939031;
        double r20939041 = r20939040 / r20939036;
        double r20939042 = r20939027 * r20939041;
        double r20939043 = r20939039 ? r20939042 : r20939037;
        double r20939044 = r20939029 ? r20939037 : r20939043;
        return r20939044;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.4
Target	0.3
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;z \lt -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -8.230194912459417e+102 or 5.118933034108623e-123 < x

   1. Initial program 0.7

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

   3. Applied div-inv 0.8

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

   4. Applied associate-*r* 1.7

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

   if -8.230194912459417e+102 < x < 5.118933034108623e-123

   1. Initial program 3.9

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

   3. Applied *-un-lft-identity 3.9

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.230194912459417 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot \left(\sin y \cdot x\right)}{z}\\ \mathbf{elif}\;x \le 5.118933034108623 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y} \cdot \left(\sin y \cdot x\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"

  :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))