Average Error: 3.6 → 1.9
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0 \le -1.2635193728348323 \cdot 10^{+275}:\\ \;\;\;\;\frac{x}{y \cdot e^{(\left(z \cdot \sqrt{a + t}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right)\right))_* \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0} \cdot y + x}\\ \end{array}\]

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 (* 2.0 (+ (fma (- (+ a (/ 5.0 6.0)) (/ (/ 2.0 3.0) t)) (- c b) (* (/ z t) (sqrt (+ t a)))) 0)) < -1.2635193728348323e+275

    1. Initial program 7.0

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

    3. Applied div-inv7.0

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

    4. Applied fma-neg4.5

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

    5. Applied simplify4.5

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

    if -1.2635193728348323e+275 < (* 2.0 (+ (fma (- (+ a (/ 5.0 6.0)) (/ (/ 2.0 3.0) t)) (- c b) (* (/ z t) (sqrt (+ t a)))) 0))

    1. Initial program 2.5

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

    3. Applied add-cube-cbrt2.5

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

    4. Applied prod-diff15.0

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

    5. Applied simplify13.7

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

    6. Applied simplify1.1

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

  4. Applied simplify1.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0 \le -1.2635193728348323 \cdot 10^{+275}:\\ \;\;\;\;\frac{x}{y \cdot e^{(\left(z \cdot \sqrt{a + t}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right)\right))_* \cdot 2.0} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{(\left(\left(a + \frac{5.0}{6.0}\right) - \frac{\frac{2.0}{3.0}}{t}\right) \cdot \left(c - b\right) + \left(\frac{z}{t} \cdot \sqrt{a + t}\right))_* \cdot 2.0} \cdot y + x}\\ \end{array}}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2019053 +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)))))))))))