Average Error: 3.7 → 1.9
Time: 22.4s
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.652084010482731 \cdot 10^{-194} \lor \neg \left(t \le -1.3295069462761203 \cdot 10^{-285}\right):\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(2.0 \cdot \left(c - b\right)\right) + \left(\sqrt{t + a} \cdot \frac{2.0 \cdot z}{t}\right))_*}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(\frac{1.3333333333333333}{t}\right) \cdot \left(b - c\right) + \left(1.6666666666666667 \cdot c\right))_*}\right) + x)_*}\\ \end{array}\]

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 t < -4.652084010482731e-194 or -1.3295069462761203e-285 < t

    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. Initial simplification1.4

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

    if -4.652084010482731e-194 < t < -1.3295069462761203e-285

    1. Initial program 8.9

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

    2. Initial simplification6.5

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

    3. Taylor expanded around 0 13.9

      \[\leadsto \frac{x}{(y \cdot \left(e^{\color{blue}{\left(1.3333333333333333 \cdot \frac{b}{t} + 1.6666666666666667 \cdot c\right) - 1.3333333333333333 \cdot \frac{c}{t}}}\right) + x)_*}\]

    4. Simplified9.8

      \[\leadsto \frac{x}{(y \cdot \left(e^{\color{blue}{(\left(\frac{1.3333333333333333}{t}\right) \cdot \left(b - c\right) + \left(1.6666666666666667 \cdot c\right))_*}}\right) + x)_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.652084010482731 \cdot 10^{-194} \lor \neg \left(t \le -1.3295069462761203 \cdot 10^{-285}\right):\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(2.0 \cdot \left(c - b\right)\right) + \left(\sqrt{t + a} \cdot \frac{2.0 \cdot z}{t}\right))_*}\right) + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(y \cdot \left(e^{(\left(\frac{1.3333333333333333}{t}\right) \cdot \left(b - c\right) + \left(1.6666666666666667 \cdot c\right))_*}\right) + x)_*}\\ \end{array}\]

Runtime

Time bar (total: 22.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.71.90.01.7-11.7%
herbie shell --seed 2018295 +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)))))))))))