Average Error: 3.6 → 1.4
Time: 2.7m
Precision: 64
Internal Precision: 320
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\frac{\sqrt{t + a}}{t}\right) \cdot z + \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(c - b\right)\right))_* - \left(b - c\right) \cdot \left(-\frac{2.0}{t \cdot 3.0}\right)\right)}} \le 1.0:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left((\left(\frac{\sqrt{t + a}}{t}\right) \cdot z + \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(c - b\right)\right))_* - \left(b - c\right) \cdot \left(-\frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right))_*}}\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 1.7

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

    3. Applied sub-neg1.7

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

    4. Applied distribute-lft-in1.7

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

    5. Applied associate--r+1.7

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

    6. Applied simplify0.3

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

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

    1. Initial program 30.2

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

    3. Applied div-inv30.2

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

    4. Applied fma-neg16.2

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

    5. Applied simplify16.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot (\left(z \cdot \sqrt{t + a}\right) \cdot \left(\frac{1}{t}\right) + \color{blue}{\left(\left(\left(\frac{5.0}{6.0} + a\right) - \frac{\frac{2.0}{t}}{3.0}\right) \cdot \left(c - b\right)\right)})_*}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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)))))))))))