Average Error: 28.4 → 28.4
Time: 9.6m
Precision: 64
Internal Precision: 384
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\frac{-\left({\left({x}^{3}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(0.1049934947 \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0072644182\right)\right)\right) + \left(x + x \cdot \left({x}^{3} \cdot \left(x \cdot 0.0424060604\right)\right)\right)\right)\right)}{-\left(\left(\left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(\left(\sqrt[3]{{\left(\left(x \cdot 0.0001789971\right) \cdot \left(x + x\right)\right)}^{3}} \cdot {\left(x \cdot x\right)}^{\left(3 + 2\right)} + 1\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0694555761 \cdot \left(x \cdot x\right) + 0.2909738639\right)\right)\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 28.4

    \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

  2. Applied simplify28.4

    \[\leadsto \color{blue}{\frac{{x}^{3} \cdot \left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(0.0001789971 \cdot x\right) \cdot x + 0.0005064034\right)\right) + \left(\left(\left(x \cdot x\right) \cdot 0.0072644182\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot 0.1049934947 + \left(\left(x \cdot 0.0424060604\right) \cdot {x}^{3} + 1\right)\right)\right) \cdot x}{\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + \left(\left(x \cdot x\right) \cdot 0.0008327945\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)}\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right) \cdot \left({\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot 0.0694555761\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left(\left(0.2909738639 \cdot x\right) \cdot {x}^{3} + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right)}}\]
  3. Using strategy rm

  4. Applied frac-2neg28.4

    \[\leadsto \color{blue}{\frac{-\left({x}^{3} \cdot \left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(0.0001789971 \cdot x\right) \cdot x + 0.0005064034\right)\right) + \left(\left(\left(x \cdot x\right) \cdot 0.0072644182\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot 0.1049934947 + \left(\left(x \cdot 0.0424060604\right) \cdot {x}^{3} + 1\right)\right)\right) \cdot x\right)}{-\left(\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + \left(\left(x \cdot x\right) \cdot 0.0008327945\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)}\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right) \cdot \left({\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot 0.0694555761\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left(\left(0.2909738639 \cdot x\right) \cdot {x}^{3} + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right)\right)}}\]

  5. Applied simplify28.4

    \[\leadsto \frac{\color{blue}{-\left({\left({x}^{3}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(0.1049934947 \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0072644182\right)\right)\right) + \left(x + x \cdot \left({x}^{3} \cdot \left(x \cdot 0.0424060604\right)\right)\right)\right)\right)}}{-\left(\left(0.0140005442 \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + \left(\left(x \cdot x\right) \cdot 0.0008327945\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)}\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right) \cdot \left({\left(x \cdot x\right)}^{\left(3 + 1\right)} \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot 0.0694555761\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left(\left(0.2909738639 \cdot x\right) \cdot {x}^{3} + 0.7715471019 \cdot \left(x \cdot x\right)\right)\right)\right)}\]

  6. Applied simplify28.4

    \[\leadsto \frac{-\left({\left({x}^{3}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(0.1049934947 \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0072644182\right)\right)\right) + \left(x + x \cdot \left({x}^{3} \cdot \left(x \cdot 0.0424060604\right)\right)\right)\right)\right)}{\color{blue}{-\left(\left(\left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(\left(\left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 2\right)} + 1\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0694555761 \cdot \left(x \cdot x\right) + 0.2909738639\right)\right)\right)}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube28.4

    \[\leadsto \frac{-\left({\left({x}^{3}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(0.1049934947 \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0072644182\right)\right)\right) + \left(x + x \cdot \left({x}^{3} \cdot \left(x \cdot 0.0424060604\right)\right)\right)\right)\right)}{-\left(\left(\left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(\left(\color{blue}{\sqrt[3]{\left(\left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.0001789971 + 0.0001789971\right)\right)}} \cdot {\left(x \cdot x\right)}^{\left(3 + 2\right)} + 1\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0694555761 \cdot \left(x \cdot x\right) + 0.2909738639\right)\right)\right)}\]

  9. Applied simplify28.4

    \[\leadsto \frac{-\left({\left({x}^{3}\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.0001789971 + 0.0005064034\right) + \left(\left(\left(0.1049934947 \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 0.0072644182\right)\right)\right) + \left(x + x \cdot \left({x}^{3} \cdot \left(x \cdot 0.0424060604\right)\right)\right)\right)\right)}{-\left(\left(\left(0.0008327945 \cdot \left(x \cdot x\right) + 0.0140005442\right) \cdot {\left(x \cdot x\right)}^{\left(3 + 1\right)} + 0.7715471019 \cdot \left(x \cdot x\right)\right) + \left(\left(\sqrt[3]{\color{blue}{{\left(\left(x \cdot 0.0001789971\right) \cdot \left(x + x\right)\right)}^{3}}} \cdot {\left(x \cdot x\right)}^{\left(3 + 2\right)} + 1\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0694555761 \cdot \left(x \cdot x\right) + 0.2909738639\right)\right)\right)}\]
  10. Removed slow pow expressions.

Runtime

Time bar (total: 9.6m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))