Average Error: 3.6 → 1.5
Time: 2.5m
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot {\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)} \le 3.5968787812686 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2.0}\right)}^{\left((z \cdot \left(\frac{1}{\frac{t}{\sqrt{t + a}}}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)}}\\ \end{array}\]

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 (pow (exp 2.0) (fma (/ z t) (sqrt (+ t a)) (- (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* t 3.0))) (- b c)))))) < 3.5968787812686e-310

    1. Initial program 1.9

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

    2. Initial simplification1.2

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

    4. Applied associate-/r/0.7

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

    5. Applied fma-neg0.2

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

    if 3.5968787812686e-310 < (* y (pow (exp 2.0) (fma (/ z t) (sqrt (+ t a)) (- (* (- (+ (/ 5.0 6.0) a) (/ 2.0 (* t 3.0))) (- b c))))))

    1. Initial program 7.6

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

    2. Initial simplification7.3

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

    4. Applied div-inv7.3

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

    5. Applied fma-neg4.8

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot {\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)} \le 3.5968787812686 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2.0}\right)}^{\left((\left(\frac{z}{t}\right) \cdot \left(\sqrt{t + a}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2.0}\right)}^{\left((z \cdot \left(\frac{1}{\frac{t}{\sqrt{t + a}}}\right) + \left(\left(-\left(b - c\right)\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{t \cdot 3.0}\right)\right))_*\right)}}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018214 +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)))))))))))