Average Error: 9.7 → 1.9
Time: 7.2m
Precision: 64
Internal Precision: 1408
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.81901803570697 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{y} \cdot e^{\log a \cdot \left(t - 1.0\right) - \left(b - \log z \cdot y\right)}\\ \mathbf{if}\;y \le 0.26022260549807996:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{{z}^{y}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot e^{\log a \cdot \left(t - 1.0\right) - \left(b - \log z \cdot y\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original9.7
Target8.8
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t \lt 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1.0\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.81901803570697e-133 or 0.26022260549807996 < y

    1. Initial program 0.6

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

    2. Applied simplify24.5

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{{z}^{y}}}}\]
    3. Using strategy rm

    4. Applied pow-to-exp24.5

      \[\leadsto \frac{x}{y} \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{\color{blue}{e^{\log z \cdot y}}}}\]

    5. Applied div-exp15.0

      \[\leadsto \frac{x}{y} \cdot \frac{{a}^{\left(t - 1.0\right)}}{\color{blue}{e^{b - \log z \cdot y}}}\]

    6. Applied pow-to-exp15.3

      \[\leadsto \frac{x}{y} \cdot \frac{\color{blue}{e^{\log a \cdot \left(t - 1.0\right)}}}{e^{b - \log z \cdot y}}\]

    7. Applied div-exp1.3

      \[\leadsto \frac{x}{y} \cdot \color{blue}{e^{\log a \cdot \left(t - 1.0\right) - \left(b - \log z \cdot y\right)}}\]

    if -4.81901803570697e-133 < y < 0.26022260549807996

    1. Initial program 20.1

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

    2. Applied simplify15.9

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{{a}^{\left(t - 1.0\right)}}{\frac{e^{b}}{{z}^{y}}}}\]
    3. Using strategy rm

    4. Applied associate-*l/2.5

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

Runtime

Time bar (total: 7.2m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1) (* y (log z))))))

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