Average Error: 4.1 → 1.6
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{(\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right))_* \cdot 2.0} \cdot y \le +\infty:\\ \;\;\;\;\frac{x}{x + e^{(\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right))_* \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + {\left(e^{2.0}\right)}^{\left((\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{t + a}\right))_*\right)} \cdot y}\\ \end{array}\]

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 (* y (exp (* 2.0 (fma (* z (sqrt (+ t a))) (/ 1 t) (* (- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0)) (- c b)))))) < +inf.0

    1. Initial program 2.9

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

    3. Applied div-inv2.9

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

    4. Applied fma-neg1.5

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

    5. Applied simplify1.5

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

    if +inf.0 < (* y (exp (* 2.0 (fma (* z (sqrt (+ t a))) (/ 1 t) (* (- (+ (/ 5.0 6.0) a) (/ (/ 2.0 t) 3.0)) (- c b))))))

    1. Initial program 62.1

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

    3. Applied add-cube-cbrt62.1

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

    4. Applied prod-diff62.5

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

    5. Applied simplify39.6

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

    6. Applied simplify9.9

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(c - b\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_* + \color{blue}{0}\right)}}\]
    7. Using strategy rm

    8. Applied exp-prod9.9

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{{\left(e^{2.0}\right)}^{\left((\left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(c - b\right) + \left(\sqrt{t + a} \cdot \frac{z}{t}\right))_* + 0\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.6

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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)))))))))))