Average Error: 2.8 → 1.7
Time: 17.0s
Precision: 64
Internal precision: 128
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
⬇
\[\begin{array}{l} \mathbf{if}\;z \le -1.82015982480053 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original2.8
Comparison0.3
Herbie1.7
\[ \begin{array}{l} \mathbf{if}\;z \lt -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{if}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if z < -1.82015982480053e+26

    1. Initial program 0.1

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

    3. Applied associate-/l* 6.6

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

    5. Applied div-inv 6.6

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

    6. Applied associate-/r* 0.1

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

    if -1.82015982480053e+26 < z

    1. Initial program 3.7

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

    3. Applied associate-/l* 2.2

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

Runtime

Time bar (total: 17.0s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(1440918070 1407420733 2964682725 2621594647 4044970290 104752456)'
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"

  :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))