Average Error: 2.8 → 1.7
Time: 14.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le 2.083597813045566445531085328327042967006 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{\frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot \frac{\sin y}{y} \le 2.083597813045566445531085328327042967006 \cdot 10^{-202}:\\
\;\;\;\;x \cdot \frac{\frac{1}{\frac{y}{\sin y}}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r412772 = x;
        double r412773 = y;
        double r412774 = sin(r412773);
        double r412775 = r412774 / r412773;
        double r412776 = r412772 * r412775;
        double r412777 = z;
        double r412778 = r412776 / r412777;
        return r412778;
}
↓
double f(double x, double y, double z) {
        double r412779 = x;
        double r412780 = y;
        double r412781 = sin(r412780);
        double r412782 = r412781 / r412780;
        double r412783 = r412779 * r412782;
        double r412784 = 2.0835978130455664e-202;
        bool r412785 = r412783 <= r412784;
        double r412786 = 1.0;
        double r412787 = r412780 / r412781;
        double r412788 = r412786 / r412787;
        double r412789 = z;
        double r412790 = r412788 / r412789;
        double r412791 = r412779 * r412790;
        double r412792 = r412786 / r412780;
        double r412793 = r412781 * r412792;
        double r412794 = r412779 * r412793;
        double r412795 = r412794 / r412789;
        double r412796 = r412785 ? r412791 : r412795;
        return r412796;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* x (/ (sin y) y)) < 2.0835978130455664e-202

    1. Initial program 4.2

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

    3. Applied *-un-lft-identity4.2

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

    4. Applied times-frac2.4

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

    5. Simplified2.4

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

    7. Applied clear-num2.4

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

    if 2.0835978130455664e-202 < (* x (/ (sin y) y))

    1. Initial program 0.2

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

    3. Applied div-inv0.2

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le 2.083597813045566445531085328327042967006 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{\frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.21737202034271466e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.44670236911381103e64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

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