Average Error: 2.0 → 2.1
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}}\right) \le 2.7703380221338586 \cdot 10^{-61}:\\ \;\;\;\;\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left({z}^{y} \cdot {a}^{\left(t - 1.0\right)}\right) \cdot e^{-b}\right)}{y}\\ \end{array}\]

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 2 regimes

  2. if (* (* (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b)))))) (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b))))))) (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b))))))) < 2.7703380221338586e-61

    1. Initial program 1.9

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

    2. Initial simplification1.4

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

    4. Applied add-cube-cbrt1.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{y} \cdot e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*}} \cdot \sqrt[3]{\frac{x}{y} \cdot e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*}}\right) \cdot \sqrt[3]{\frac{x}{y} \cdot e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*}}}\]

    if 2.7703380221338586e-61 < (* (* (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b)))))) (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b))))))) (cbrt (* (/ x y) (exp (fma (log z) y (fma (log a) (- t 1.0) (- b)))))))

    1. Initial program 2.4

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

    3. Applied sub-neg2.4

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

    4. Applied exp-sum8.2

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

    5. Simplified6.2

      \[\leadsto \frac{x \cdot \left(\color{blue}{\left({z}^{y} \cdot {a}^{\left(t - 1.0\right)}\right)} \cdot e^{-b}\right)}{y}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}}\right) \le 2.7703380221338586 \cdot 10^{-61}:\\ \;\;\;\;\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}} \cdot \sqrt[3]{e^{(\left(\log z\right) \cdot y + \left((\left(\log a\right) \cdot \left(t - 1.0\right) + \left(-b\right))_*\right))_*} \cdot \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left({z}^{y} \cdot {a}^{\left(t - 1.0\right)}\right) \cdot e^{-b}\right)}{y}\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018214 +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))