Average Error: 2.7 → 1.8
Time: 14.7s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.883506706037952 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.883506706037952 \cdot 10^{-64}:\\
\;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r25886652 = x;
        double r25886653 = y;
        double r25886654 = sin(r25886653);
        double r25886655 = r25886654 / r25886653;
        double r25886656 = r25886652 * r25886655;
        double r25886657 = z;
        double r25886658 = r25886656 / r25886657;
        return r25886658;
}
↓
double f(double x, double y, double z) {
        double r25886659 = x;
        double r25886660 = 3.883506706037952e-64;
        bool r25886661 = r25886659 <= r25886660;
        double r25886662 = z;
        double r25886663 = y;
        double r25886664 = sin(r25886663);
        double r25886665 = r25886664 / r25886663;
        double r25886666 = r25886662 / r25886665;
        double r25886667 = r25886659 / r25886666;
        double r25886668 = 1.0;
        double r25886669 = r25886659 * r25886665;
        double r25886670 = r25886662 / r25886669;
        double r25886671 = r25886668 / r25886670;
        double r25886672 = r25886661 ? r25886667 : r25886671;
        return r25886672;
}

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
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.883506706037952e-64

    1. Initial program 3.7

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

    3. Applied associate-/l*2.1

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

    if 3.883506706037952e-64 < x

    1. Initial program 0.5

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

    3. Applied clear-num0.9

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.883506706037952 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))