Average Error: 28.7 → 28.5
Time: 49.5s
Precision: 64
Internal Precision: 384
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;(x \cdot \left((e^{(e^{\log_* (1 + \log_* (1 + \left(0.265709700396151 \cdot {x}^{4} - \left(x \cdot x\right) \cdot 0.6665536072\right)))} - 1)^*} - 1)^*\right) + x)_* \le -3.2953524053841425 \cdot 10^{+188}:\\ \;\;\;\;(x \cdot \left(\left((1.1253816727886299 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{4}}\right) + \left(\frac{\frac{-0.6665536072}{x \cdot x}}{\frac{{x}^{8}}{0.5980496542159722}}\right))_* + (0.07945379722187383 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{12}}\right) + \left(\frac{-0.6665536072}{x \cdot x}\right))_*\right) - \left(\left((0.3986321543023689 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{6}}\right) + \left(\frac{\frac{-0.6665536072}{x \cdot x}}{\frac{x \cdot x}{1.5002544269480627}}\right))_* + (\left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{12} \cdot \left(-0.6665536072 \cdot -0.6665536072\right)}\right) \cdot 0.03530082244230616 + 1)_*\right) + \frac{\frac{-0.6665536072}{x \cdot x}}{-0.6665536072 \cdot -0.6665536072} \cdot \left(\frac{\frac{1}{2}}{{x}^{4}} + \frac{0.26570970039615094}{{x}^{8}}\right)\right)\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0001789971 + \left((0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + \left((0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \left((0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((0.1049934947 \cdot \left(x \cdot x\right) + 1)_*\right))_*\right))_*\right))_*\right))_*}{(\left(\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0001789971 + 0.0001789971\right) + \left(\left((\left(0.2909738639 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(0.7715471019 \cdot \left(x \cdot x\right)\right))_* + (\left({x}^{3} \cdot {x}^{3}\right) \cdot 0.0694555761 + 1)_*\right) + (\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0008327945 + \left(\left(0.0140005442 \cdot \left(x \cdot x\right)\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right))_*\right))_*} \cdot x\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (fma x (expm1 (expm1 (log1p (log1p (- (* 0.265709700396151 (pow x 4)) (* (* x x) 0.6665536072)))))) x) < -3.2953524053841425e+188

    1. Initial program 63.4

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

    2. Applied simplify63.4

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

    3. Taylor expanded around 0 62.3

      \[\leadsto \color{blue}{\left(\left(1 + 0.265709700396151 \cdot {x}^{4}\right) - 0.6665536072 \cdot {x}^{2}\right)} \cdot x\]

    4. Applied simplify62.3

      \[\leadsto \color{blue}{(x \cdot \left(0.265709700396151 \cdot {x}^{4} - \left(x \cdot x\right) \cdot 0.6665536072\right) + x)_*}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u62.3

      \[\leadsto (x \cdot \color{blue}{\left((e^{\log_* (1 + \left(0.265709700396151 \cdot {x}^{4} - \left(x \cdot x\right) \cdot 0.6665536072\right))} - 1)^*\right)} + x)_*\]

    7. Taylor expanded around inf 63.4

      \[\leadsto (x \cdot \color{blue}{\left(\left(1.1253816727886299 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{x}^{4}} + \left(0.07945379722187383 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{x}^{12}} + \left(e^{\log \left(-0.6665536072\right) - 2 \cdot \log x} + 0.5980496542159722 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{x}^{8}}\right)\right)\right) - \left(0.26570970039615094 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{-0.6665536072}^{2} \cdot {x}^{8}} + \left(\frac{1}{2} \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{-0.6665536072}^{2} \cdot {x}^{4}} + \left(1.5002544269480627 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{x}^{2}} + \left(0.3986321543023689 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{x}^{6}} + \left(1 + 0.03530082244230616 \cdot \frac{e^{\log \left(-0.6665536072\right) - 2 \cdot \log x}}{{-0.6665536072}^{2} \cdot {x}^{12}}\right)\right)\right)\right)\right)\right)} + x)_*\]

    8. Applied simplify61.5

      \[\leadsto \color{blue}{(x \cdot \left(\left((1.1253816727886299 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{4}}\right) + \left(\frac{\frac{-0.6665536072}{x \cdot x}}{\frac{{x}^{8}}{0.5980496542159722}}\right))_* + (0.07945379722187383 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{12}}\right) + \left(\frac{-0.6665536072}{x \cdot x}\right))_*\right) - \left(\left((0.3986321543023689 \cdot \left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{6}}\right) + \left(\frac{\frac{-0.6665536072}{x \cdot x}}{\frac{x \cdot x}{1.5002544269480627}}\right))_* + (\left(\frac{\frac{-0.6665536072}{x \cdot x}}{{x}^{12} \cdot \left(-0.6665536072 \cdot -0.6665536072\right)}\right) \cdot 0.03530082244230616 + 1)_*\right) + \frac{\frac{-0.6665536072}{x \cdot x}}{-0.6665536072 \cdot -0.6665536072} \cdot \left(\frac{\frac{1}{2}}{{x}^{4}} + \frac{0.26570970039615094}{{x}^{8}}\right)\right)\right) + x)_*}\]

    if -3.2953524053841425e+188 < (fma x (expm1 (expm1 (log1p (log1p (- (* 0.265709700396151 (pow x 4)) (* (* x x) 0.6665536072)))))) x)

    1. Initial program 25.4

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

    2. Applied simplify25.4

      \[\leadsto \color{blue}{\frac{(\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0001789971 + \left((0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right) + \left((0.0072644182 \cdot \left({x}^{3} \cdot {x}^{3}\right) + \left((0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left((0.1049934947 \cdot \left(x \cdot x\right) + 1)_*\right))_*\right))_*\right))_*\right))_*}{(\left(\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.0001789971 + 0.0001789971\right) + \left(\left((\left(0.2909738639 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(0.7715471019 \cdot \left(x \cdot x\right)\right))_* + (\left({x}^{3} \cdot {x}^{3}\right) \cdot 0.0694555761 + 1)_*\right) + (\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0008327945 + \left(\left(0.0140005442 \cdot \left(x \cdot x\right)\right) \cdot \left({x}^{3} \cdot {x}^{3}\right)\right))_*\right))_*} \cdot x}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 49.5s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(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))