Average Error: 2.7 → 1.9
Time: 7.7s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.210199914794309130085332727707594995869 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{\frac{\sin y}{y}}}{x}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.210199914794309130085332727707594995869 \cdot 10^{-88}:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{z}{\frac{\sin y}{y}}}{x}}\\

\end{array}
double f(double x, double y, double z) {
        double r566394 = x;
        double r566395 = y;
        double r566396 = sin(r566395);
        double r566397 = r566396 / r566395;
        double r566398 = r566394 * r566397;
        double r566399 = z;
        double r566400 = r566398 / r566399;
        return r566400;
}
↓
double f(double x, double y, double z) {
        double r566401 = z;
        double r566402 = -8.210199914794309e-88;
        bool r566403 = r566401 <= r566402;
        double r566404 = x;
        double r566405 = r566404 / r566401;
        double r566406 = 1.0;
        double r566407 = y;
        double r566408 = sin(r566407);
        double r566409 = r566408 / r566407;
        double r566410 = r566406 / r566409;
        double r566411 = r566405 / r566410;
        double r566412 = r566401 / r566409;
        double r566413 = r566412 / r566404;
        double r566414 = r566406 / r566413;
        double r566415 = r566403 ? r566411 : r566414;
        return r566415;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.7
Target0.3
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.210199914794309e-88

    1. Initial program 0.4

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

    3. Applied associate-/l*4.0

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

    5. Applied div-inv4.0

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

    6. Applied associate-/r*0.4

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

    if -8.210199914794309e-88 < z

    1. Initial program 3.9

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

    3. Applied associate-/l*2.2

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

    5. Applied clear-num2.7

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.210199914794309130085332727707594995869 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{\frac{\sin y}{y}}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))