Average Error: 1.9 → 1.4
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -0.4958696554030931 \lor \neg \left(y \le 0.00012170764008272702\right):\\ \;\;\;\;\frac{x}{\frac{y}{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y}{e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right)}}\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

  2. if y < -0.4958696554030931 or 0.00012170764008272702 < y

    1. Initial program 0.0

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

    3. Applied associate-/l*0.0

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

    5. Applied *-un-lft-identity0.0

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

    6. Applied exp-prod0.0

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

    7. Simplified0.0

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

    if -0.4958696554030931 < y < 0.00012170764008272702

    1. Initial program 3.7

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

    3. Applied associate-/l*3.8

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

    5. Applied *-un-lft-identity3.8

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

    6. Applied exp-prod3.8

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

    7. Simplified3.8

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

    8. Taylor expanded around inf 3.8

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

    9. Simplified2.4

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

    11. Applied div-inv2.6

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{y}{\left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right) \cdot e^{(t \cdot \left(\log a\right) + \left(-b\right))_*}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -0.4958696554030931 \lor \neg \left(y \le 0.00012170764008272702\right):\\ \;\;\;\;\frac{x}{\frac{y}{{e}^{\left(\left(\log a \cdot \left(t - 1.0\right) + y \cdot \log z\right) - b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{y}{e^{(t \cdot \left(\log a\right) + \left(-b\right))_*} \cdot \left({a}^{\left(-1.0\right)} \cdot {z}^{y}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018258 +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))