Average Error: 4.1 → 3.5
Time: 3.7m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.078632103487414 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{x + \left(\left(\sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}} \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right) \cdot y}\\ \mathbf{if}\;t \le 1.2647796154410053 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{\left(\sqrt{a + t} \cdot \left(\frac{5.0}{6.0} - a\right)\right) \cdot t - \left(\left(\frac{5.0}{6.0} - a\right) \cdot \frac{b - c}{\frac{z}{t}}\right) \cdot \left(\frac{t \cdot 5.0}{6.0} - \left(0.6666666666666666 - a \cdot t\right)\right)}{\frac{t}{z} \cdot \left(\left(\frac{5.0}{6.0} - a\right) \cdot t\right)}\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(\left(\sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}} \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right) \cdot y}\\ \end{array}\]

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 t < -3.078632103487414e-89 or 1.2647796154410053e-55 < t

    1. Initial program 2.6

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

    3. Applied add-log-exp5.7

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

    4. Taylor expanded around 0 5.7

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

    5. Applied simplify0.5

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

    7. Applied add-cube-cbrt0.5

      \[\leadsto \frac{x}{x + \color{blue}{\left(\left(\sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}} \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right) \cdot \sqrt[3]{{\left(e^{2.0}\right)}^{\left(\frac{\sqrt{t + a}}{\frac{t}{z}} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{0.6666666666666666}{t}\right) \cdot \left(b - c\right)\right)}}\right)} \cdot y}\]

    if -3.078632103487414e-89 < t < 1.2647796154410053e-55

    1. Initial program 6.3

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

    3. Applied add-log-exp17.5

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

    4. Taylor expanded around 0 17.5

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

    5. Applied simplify6.9

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

    7. Applied flip-+10.1

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

    8. Applied frac-sub10.1

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

    9. Applied associate-*l/10.1

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

    10. Applied frac-sub13.0

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

    11. Applied simplify8.2

      \[\leadsto \frac{x}{x + {\left(e^{2.0}\right)}^{\left(\frac{\color{blue}{\left(\sqrt{a + t} \cdot \left(\frac{5.0}{6.0} - a\right)\right) \cdot t - \left(\left(\frac{5.0}{6.0} - a\right) \cdot \frac{b - c}{\frac{z}{t}}\right) \cdot \left(\frac{t \cdot 5.0}{6.0} - \left(0.6666666666666666 - a \cdot t\right)\right)}}{\frac{t}{z} \cdot \left(\left(\frac{5.0}{6.0} - a\right) \cdot t\right)}\right)} \cdot y}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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)))))))))))