Average Error: 1.8 → 0.9
Time: 2.8m
Precision: 64
Internal Precision: 384
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\sqrt[3]{{\left(e^{b}\right)}^{3} \cdot {\left(\frac{y}{x}\right)}^{3}} \cdot {a}^{\left(-t\right)}} \le -2.2793740129648608 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{x}{{a}^{1.0}}}{y} \cdot \left(t \cdot \log a + y \cdot \log z\right) + \frac{\frac{x}{{a}^{1.0}}}{y}\\ \mathbf{if}\;\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\sqrt[3]{{\left(e^{b}\right)}^{3} \cdot {\left(\frac{y}{x}\right)}^{3}} \cdot {a}^{\left(-t\right)}} \le +\infty:\\ \;\;\;\;\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\frac{\frac{y}{x} \cdot e^{b}}{{a}^{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if (/ (* (pow z y) (pow a (- 1.0))) (* (cbrt (* (pow (exp b) 3) (pow (/ y x) 3))) (pow a (- t)))) < -2.2793740129648608e+135

    1. Initial program 7.4

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

    2. Taylor expanded around 0 10.8

      \[\leadsto \frac{x \cdot \color{blue}{\left(\log a \cdot \left(e^{-1.0 \cdot \log a} \cdot t\right) + \left(y \cdot \left(e^{-1.0 \cdot \log a} \cdot \log z\right) + e^{-1.0 \cdot \log a}\right)\right)}}{y}\]

    3. Applied simplify6.3

      \[\leadsto \color{blue}{\frac{\frac{x}{{a}^{1.0}}}{y} \cdot \left(t \cdot \log a + y \cdot \log z\right) + \frac{\frac{x}{{a}^{1.0}}}{y}}\]

    if -2.2793740129648608e+135 < (/ (* (pow z y) (pow a (- 1.0))) (* (cbrt (* (pow (exp b) 3) (pow (/ y x) 3))) (pow a (- t)))) < +inf.0

    1. Initial program 2.9

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

    2. Taylor expanded around inf 2.9

      \[\leadsto \frac{\color{blue}{e^{1.0 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(b + t \cdot \log \left(\frac{1}{a}\right)\right)\right)} \cdot x}}{y}\]

    3. Applied simplify1.2

      \[\leadsto \color{blue}{\frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\left(\frac{y}{x} \cdot e^{b}\right) \cdot {a}^{\left(-t\right)}}}\]
    4. Using strategy rm

    5. Applied pow-neg1.2

      \[\leadsto \frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\left(\frac{y}{x} \cdot e^{b}\right) \cdot \color{blue}{\frac{1}{{a}^{t}}}}\]

    6. Applied un-div-inv1.2

      \[\leadsto \frac{{z}^{y} \cdot {a}^{\left(-1.0\right)}}{\color{blue}{\frac{\frac{y}{x} \cdot e^{b}}{{a}^{t}}}}\]

    if +inf.0 < (/ (* (pow z y) (pow a (- 1.0))) (* (cbrt (* (pow (exp b) 3) (pow (/ y x) 3))) (pow a (- t))))

    1. Initial program 0.1

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

      \[\leadsto \frac{x \cdot e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}}{y}\]

    4. Applied exp-prod0.1

      \[\leadsto \frac{x \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}}{y}\]

    5. Applied simplify0.1

      \[\leadsto \frac{x \cdot {\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}{y}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))