Average Error: 20.7 → 1.9
Time: 3.4m
Precision: 64
Internal precision: 128
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
⬇
\[\begin{array}{l} \mathbf{if}\;t \le -2.4368284028755833 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(0.8333333333333334 \cdot c + c \cdot a\right) - b \cdot a\right)}}\\ \mathbf{if}\;t \le 2.3351005256458113 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\frac{\frac{2.0}{t}}{3.0} + \left(\frac{5.0}{6.0} + a\right)\right) \cdot \left(z \cdot \sqrt{a + t} - \left(\left(b - c\right) \cdot t\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right)\right)}{t \cdot \left(\left(a + \frac{5.0}{6.0}\right) + \frac{2.0}{t \cdot 3.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

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

Target

Original20.7
Comparison2.5
Herbie1.9
\[ \begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{if}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left({t}^2 \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if t < -2.4368284028755833e+34

    1. Initial program 55.3

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

    2. Applied taylor 1.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(0.8333333333333334 \cdot c + c \cdot a\right) - b \cdot a\right)}}\]

    3. Taylor expanded around inf 1.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\left(0.8333333333333334 \cdot c + c \cdot a\right) - b \cdot a\right)}}}\]

    if -2.4368284028755833e+34 < t < 2.3351005256458113e-142

    1. Initial program 6.7

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

    3. Applied flip-- 20.2

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

    4. Applied associate-*r/ 20.4

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

    5. Applied frac-sub 21.3

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

    6. Applied simplify 1.7

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

    if 2.3351005256458113e-142 < t

    1. Initial program 2.3

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 3.4m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2901879629 3206232249 1557822768 2386024429 645228315 4049442648)'
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (sqr t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

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