Average Error: 1.9 → 0.7
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{x}{y} \cdot e^{-b}\right) \cdot \frac{{\left(\frac{1}{z}\right)}^{\left(-y\right)}}{{\left(\frac{1}{a}\right)}^{\left(t - 1.0\right)}} \le -2.7435541320658977 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{\left({z}^{y} \cdot {a}^{t}\right) \cdot x}{{a}^{1.0} \cdot e^{b}}}{y}\\ \mathbf{if}\;\left(\frac{x}{y} \cdot e^{-b}\right) \cdot \frac{{\left(\frac{1}{z}\right)}^{\left(-y\right)}}{{\left(\frac{1}{a}\right)}^{\left(t - 1.0\right)}} \le +\infty:\\ \;\;\;\;\left(\frac{x}{y} \cdot e^{-b}\right) \cdot \frac{{\left(\frac{1}{z}\right)}^{\left(-y\right)}}{{\left(\frac{1}{a}\right)}^{\left(t - 1.0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\\ \end{array}\]

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 (* (* (/ x y) (exp (- b))) (/ (pow (/ 1 z) (- y)) (pow (/ 1 a) (- t 1.0)))) < -2.7435541320658977e-95

    1. Initial program 6.6

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

    3. Applied sub-neg6.6

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

    4. Applied exp-sum6.6

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

    5. Applied simplify1.8

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

    7. Applied exp-neg1.8

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

    8. Applied pow-sub1.6

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

    9. Applied associate-*r/1.6

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

    10. Applied frac-times1.6

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

    11. Applied associate-*r/1.5

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

    12. Applied simplify1.5

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

    if -2.7435541320658977e-95 < (* (* (/ x y) (exp (- b))) (/ (pow (/ 1 z) (- y)) (pow (/ 1 a) (- t 1.0)))) < +inf.0

    1. Initial program 2.5

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

    3. Applied sub-neg2.5

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

    4. Applied exp-sum2.5

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

    5. Applied simplify1.9

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

    6. Taylor expanded around inf 2.5

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

    7. Applied simplify1.0

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

    if +inf.0 < (* (* (/ x y) (exp (- b))) (/ (pow (/ 1 z) (- y)) (pow (/ 1 a) (- t 1.0))))

    1. Initial program 0

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(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))