Average Error: 2.8 → 0.2
Time: 19.1s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5593196487055916383862784:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{elif}\;z \le 0.001397275382843652952874480277500879310537:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5593196487055916383862784:\\
\;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\

\mathbf{elif}\;z \le 0.001397275382843652952874480277500879310537:\\
\;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r19723653 = x;
        double r19723654 = y;
        double r19723655 = sin(r19723654);
        double r19723656 = r19723655 / r19723654;
        double r19723657 = r19723653 * r19723656;
        double r19723658 = z;
        double r19723659 = r19723657 / r19723658;
        return r19723659;
}
↓
double f(double x, double y, double z) {
        double r19723660 = z;
        double r19723661 = -5.593196487055916e+24;
        bool r19723662 = r19723660 <= r19723661;
        double r19723663 = x;
        double r19723664 = 1.0;
        double r19723665 = y;
        double r19723666 = r19723664 / r19723665;
        double r19723667 = sin(r19723665);
        double r19723668 = r19723664 / r19723667;
        double r19723669 = r19723666 / r19723668;
        double r19723670 = r19723663 * r19723669;
        double r19723671 = r19723670 / r19723660;
        double r19723672 = 0.001397275382843653;
        bool r19723673 = r19723660 <= r19723672;
        double r19723674 = r19723667 / r19723665;
        double r19723675 = r19723660 / r19723674;
        double r19723676 = r19723663 / r19723675;
        double r19723677 = r19723673 ? r19723676 : r19723671;
        double r19723678 = r19723662 ? r19723671 : r19723677;
        return r19723678;
}

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

Original2.8
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.593196487055916e+24 or 0.001397275382843653 < z

    1. Initial program 0.1

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

    3. Applied clear-num0.1

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

    5. Applied div-inv0.2

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

    6. Applied associate-/r*0.2

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

    if -5.593196487055916e+24 < z < 0.001397275382843653

    1. Initial program 5.7

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5593196487055916383862784:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{elif}\;z \le 0.001397275382843652952874480277500879310537:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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 (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))