Average Error: 3.8 → 2.5
Time: 9.2m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(z \cdot \sqrt{a + t} - \frac{\left(b - c\right) \cdot t}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right)} \cdot \left(\left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)\right)\right)}} \le 1.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(z \cdot \sqrt{a + t} - \frac{\left(b - c\right) \cdot t}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right)} \cdot \left(\left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{z \cdot \sqrt{a + t}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \log \left(e^{\frac{2.0}{3.0 \cdot t}}\right)\right)\right) \cdot 2.0} + x}\\ \end{array}\]

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 (/ x (+ x (* y (exp (* 2.0 (* (/ 1 t) (- (* 1 (* z (sqrt (+ t a)))) (* (/ (* (- b c) t) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- a (/ 5.0 6.0)) (- (* (* 3.0 t) (+ (/ 5.0 6.0) a)) 2.0)))))))))) < 1.0

    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 flip-+3.7

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

    4. Applied frac-sub3.8

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

    5. Applied associate-*r/3.9

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

    6. Applied frac-sub16.0

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

    8. Applied *-un-lft-identity16.0

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

    9. Applied times-frac7.2

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

    10. Applied simplify0.7

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

    if 1.0 < (/ x (+ x (* y (exp (* 2.0 (* (/ 1 t) (- (* 1 (* z (sqrt (+ t a)))) (* (/ (* (- b c) t) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- a (/ 5.0 6.0)) (- (* (* 3.0 t) (+ (/ 5.0 6.0) a)) 2.0))))))))))

    1. Initial program 10.3

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

    3. Applied add-log-exp11.7

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

  4. Applied simplify2.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(z \cdot \sqrt{a + t} - \frac{\left(b - c\right) \cdot t}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right)} \cdot \left(\left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)\right)\right)}} \le 1.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{1}{t} \cdot \left(z \cdot \sqrt{a + t} - \frac{\left(b - c\right) \cdot t}{\left(a - \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right)} \cdot \left(\left(\left(a + \frac{5.0}{6.0}\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{\left(\frac{z \cdot \sqrt{a + t}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \log \left(e^{\frac{2.0}{3.0 \cdot t}}\right)\right)\right) \cdot 2.0} + x}\\ \end{array}}\]

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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)))))))))))