Average Error: 2.7 → 0.4
Time: 7.6s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.5463407620259075 \cdot 10^{83} \lor \neg \left(z \le 1.00828897081930849 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.5463407620259075 \cdot 10^{83} \lor \neg \left(z \le 1.00828897081930849 \cdot 10^{-64}\right):\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r1546 = x;
        double r1547 = y;
        double r1548 = sin(r1547);
        double r1549 = r1548 / r1547;
        double r1550 = r1546 * r1549;
        double r1551 = z;
        double r1552 = r1550 / r1551;
        return r1552;
}
↓
double f(double x, double y, double z) {
        double r1553 = z;
        double r1554 = -8.546340762025908e+83;
        bool r1555 = r1553 <= r1554;
        double r1556 = 1.0082889708193085e-64;
        bool r1557 = r1553 <= r1556;
        double r1558 = !r1557;
        bool r1559 = r1555 || r1558;
        double r1560 = x;
        double r1561 = r1560 / r1553;
        double r1562 = 1.0;
        double r1563 = y;
        double r1564 = sin(r1563);
        double r1565 = r1564 / r1563;
        double r1566 = r1562 / r1565;
        double r1567 = r1561 / r1566;
        double r1568 = r1553 / r1565;
        double r1569 = r1560 / r1568;
        double r1570 = r1559 ? r1567 : r1569;
        return r1570;
}

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.546340762025908e+83 or 1.0082889708193085e-64 < z

    1. Initial program 0.2

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

    3. Applied clear-num0.9

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

    5. Applied div-inv1.0

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

    6. Applied associate-/r*0.4

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

    8. Applied *-un-lft-identity0.4

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

    9. Applied times-frac0.5

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

    10. Applied associate-/r*0.5

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

    11. Simplified0.3

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

    if -8.546340762025908e+83 < z < 1.0082889708193085e-64

    1. Initial program 5.4

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

    3. Applied clear-num5.6

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

    5. Applied div-inv6.1

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

    6. Applied associate-/r*6.0

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

    8. Applied *-un-lft-identity6.0

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

    9. Applied *-un-lft-identity6.0

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

    10. Applied times-frac6.0

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

    11. Applied associate-/l*6.2

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

    12. Simplified5.6

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

    14. Applied *-un-lft-identity5.6

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

    15. Applied times-frac1.0

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

    16. Applied associate-/r*0.6

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

    17. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.5463407620259075 \cdot 10^{83} \lor \neg \left(z \le 1.00828897081930849 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))