Average Error: 3.7 → 2.7
Time: 3.5m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} \le -inf.0:\\ \;\;\;\;\frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(b - c\right) \cdot \left(\frac{5.0}{6.0} + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right)} \cdot y}\\ \mathbf{if}\;\frac{z \cdot \sqrt{t + a}}{t} \le 1.0083479312050176 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \log \left(e^{\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(b - c\right) \cdot \left(\frac{5.0}{6.0} + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right)} \cdot y}\\ \end{array}\]

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 (/ (* z (sqrt (+ t a))) t) or 1.0083479312050176e+174 < (/ (* z (sqrt (+ t a))) t)

    1. Initial program 13.8

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

    3. Applied add-log-exp13.8

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

    4. Taylor expanded around 0 13.8

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \log \left(e^{\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\frac{0.6666666666666666}{t}}\right)}\right)}}\]

    5. Applied simplify9.9

      \[\leadsto \color{blue}{\frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(b - c\right) \cdot \left(\frac{5.0}{6.0} + \left(a - \frac{0.6666666666666666}{t}\right)\right)\right)} \cdot y}}\]

    if (/ (* z (sqrt (+ t a))) t) < 1.0083479312050176e+174

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(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)))))))))))