Average Error: 4.1 → 5.7
Time: 2.9m
Precision: 64
Internal Precision: 384
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.159894663220627 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}} \cdot \sqrt{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}}\\ \mathbf{if}\;t \le 3.791128916785116 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{(\left(e^{\frac{(2.0 \cdot \left(\frac{5.0}{6.0} + a\right) + \left(\frac{2.0 \cdot 2.0}{3.0 \cdot t}\right))_*}{(t \cdot \left(\frac{5.0}{6.0} + a\right) + \left(\frac{2.0}{3.0}\right))_*} \cdot \left(\sqrt{t + a} \cdot z - (\left(0.8333333333333334 \cdot b\right) \cdot t + \left(0.6666666666666666 \cdot \left(c - b\right)\right))_*\right)}\right) \cdot y + x)_*}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*})} - 1)^*\\ \end{array}\]

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 3 regimes

  2. if t < -4.159894663220627e-77

    1. Initial program 3.5

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

    2. Applied simplify1.8

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

    3. Taylor expanded around inf 5.9

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

    4. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt1.7

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

    if -4.159894663220627e-77 < t < 3.791128916785116e+41

    1. Initial program 5.0

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

    3. Applied flip--14.0

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

    4. Applied associate-*r/14.4

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

    5. Applied frac-sub15.2

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

    6. Applied simplify1.0

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

    7. Applied simplify1.0

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

    8. Taylor expanded around 0 6.0

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(\frac{2.0}{3.0 \cdot t} + \left(\frac{5.0}{6.0} + a\right)\right) \cdot \left(z \cdot \sqrt{t + a} - \color{blue}{\left(\left(0.6666666666666666 \cdot c + 0.8333333333333334 \cdot \left(b \cdot t\right)\right) - 0.6666666666666666 \cdot b\right)}\right)}{(t \cdot \left(a + \frac{5.0}{6.0}\right) + \left(\frac{2.0}{3.0}\right))_*}}}\]

    9. Applied simplify6.0

      \[\leadsto \color{blue}{\frac{x}{(\left(e^{\frac{(2.0 \cdot \left(\frac{5.0}{6.0} + a\right) + \left(\frac{2.0 \cdot 2.0}{3.0 \cdot t}\right))_*}{(t \cdot \left(\frac{5.0}{6.0} + a\right) + \left(\frac{2.0}{3.0}\right))_*} \cdot \left(\sqrt{t + a} \cdot z - (\left(0.8333333333333334 \cdot b\right) \cdot t + \left(0.6666666666666666 \cdot \left(c - b\right)\right))_*\right)}\right) \cdot y + x)_*}}\]

    if 3.791128916785116e+41 < t

    1. Initial program 3.0

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

    2. Applied simplify0.1

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

    3. Taylor expanded around inf 7.5

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

    4. Applied simplify6.5

      \[\leadsto \color{blue}{\frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*}}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u6.5

      \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{x}{(y \cdot \left({\left(e^{2.0}\right)}^{\left((0.8333333333333334 \cdot c + \left((\left(\sqrt{t + a}\right) \cdot \left(\frac{z}{t}\right) + \left(\left(-a\right) \cdot \left(b - c\right)\right))_*\right))_*\right)}\right) + x)_*})} - 1)^*}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.9m)Debug logProfile

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