Average Error: 3.8 → 1.3
Time: 3.3m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.836242278030348 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \log_* (1 + (e^{(\left(c - b\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) + \left(\frac{\sqrt{t + a}}{\frac{t}{z}}\right))_*} - 1)^*)}}\\ \mathbf{if}\;z \le 6.836077009857302 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \log_* (1 + (e^{(\left(c - b\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) + \left(\frac{\sqrt{t + a}}{\frac{t}{z}}\right))_*} - 1)^*)}}\\ \end{array}\]

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 < -1.836242278030348e+174 or 6.836077009857302e+220 < z

    1. Initial program 12.6

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

    3. Applied log1p-expm1-u12.6

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

    4. Applied simplify3.7

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

    if -1.836242278030348e+174 < z < 6.836077009857302e+220

    1. Initial program 1.8

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

    3. Applied div-inv1.8

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

    4. Applied fma-neg0.7

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

    5. Applied simplify0.7

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \color{blue}{\left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right)})_*}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +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)))))))))))