Average Error: 3.5 → 2.0
Time: 1.1m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.0575649803268776 \cdot 10^{-227}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \log \left(e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(-\left(b - c\right)\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_*}\right)}}\\ \end{array}\]

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

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if y < -1.0575649803268776e-227

    1. Initial program 3.4

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
    2. Using strategy rm

    3. Applied div-inv3.4

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\left(z \cdot \sqrt{t + a}\right) \cdot \frac{1}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    4. Applied fma-neg2.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{(\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(-\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*}}}\]

    5. Applied simplify2.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \color{blue}{\left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right)\right)})_*}}\]

    if -1.0575649803268776e-227 < y

    1. Initial program 3.7

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
    2. Using strategy rm

    3. Applied add-log-exp8.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}\right)}\right)}}\]

    4. Applied add-log-exp16.8

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\log \left(e^{\frac{z \cdot \sqrt{t + a}}{t}}\right)} - \log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}\right)\right)}}\]

    5. Applied diff-log16.8

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\log \left(\frac{e^{\frac{z \cdot \sqrt{t + a}}{t}}}{e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}}\right)}}}\]

    6. Applied simplify1.7

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \log \color{blue}{\left(e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(-\left(b - c\right)\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_*}\right)}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))