Average Error: 33.2 → 10.8
Time: 53.3s
Precision: 64
Internal precision: 128
\[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
⬇
\[\begin{array}{l} \mathbf{if}\;z \le -5.9430640354654945 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\\ \mathbf{if}\;z \le 2.3218028994593566 \cdot 10^{-103}:\\ \;\;\;\;\left(x - \frac{\log 1.0}{t}\right) - \left(\frac{z}{t} \cdot y\right) \cdot \left(0.5 \cdot z + 1.0\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\frac{1}{2} \cdot \left(y \cdot {z}^2\right) + \left(1.0 + y \cdot z\right)\right)}{t}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.2
Comparison23.5
Herbie10.8
\[ \begin{array}{l} \mathbf{if}\;z \lt -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{{z}^2}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2.0}{z}}{{z}^2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1.0 + z \cdot y\right)}{t}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if z < -5.9430640354654945e+19

    1. Initial program 11.8

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    if -5.9430640354654945e+19 < z < 2.3218028994593566e-103

    1. Initial program 33.6

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Applied taylor 1.7

      \[\leadsto x - \left(1.0 \cdot \frac{y \cdot z}{t} + \left(0.5 \cdot \frac{y \cdot {z}^2}{t} + \frac{\log 1.0}{t}\right)\right)\]

    3. Taylor expanded around 0 1.7

      \[\leadsto x - \color{blue}{\left(1.0 \cdot \frac{y \cdot z}{t} + \left(0.5 \cdot \frac{y \cdot {z}^2}{t} + \frac{\log 1.0}{t}\right)\right)}\]

    4. Applied simplify 5.9

      \[\leadsto \color{blue}{\left(x - \frac{\log 1.0}{t}\right) - z \cdot \left(\frac{y}{t} \cdot \left(z \cdot 0.5 + 1.0\right)\right)}\]

    5. Applied taylor 5.9

      \[\leadsto \left(x - \frac{\log 1.0}{t}\right) - z \cdot \left(1.0 \cdot \frac{y}{t} + 0.5 \cdot \frac{y \cdot z}{t}\right)\]

    6. Taylor expanded around 0 5.9

      \[\leadsto \left(x - \frac{\log 1.0}{t}\right) - z \cdot \color{blue}{\left(1.0 \cdot \frac{y}{t} + 0.5 \cdot \frac{y \cdot z}{t}\right)}\]

    7. Applied simplify 1.1

      \[\leadsto \color{blue}{\left(x - \frac{\log 1.0}{t}\right) - \left(\frac{z}{t} \cdot y\right) \cdot \left(0.5 \cdot z + 1.0\right)}\]

    if 2.3218028994593566e-103 < z

    1. Initial program 50.6

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Applied taylor 29.5

      \[\leadsto x - \frac{\log \left(\frac{1}{2} \cdot \left(y \cdot {z}^2\right) + \left(1.0 + y \cdot z\right)\right)}{t}\]

    3. Taylor expanded around 0 29.5

      \[\leadsto x - \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \left(y \cdot {z}^2\right) + \left(1.0 + y \cdot z\right)\right)}}{t}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 53.3s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(3666465132 719130484 1799563395 3905432248 3804154138 1071404319)'
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"

  :target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (sqr z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (sqr z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))