Average Error: 13.8 → 1.9
Time: 2.7m
Precision: 64
Internal Precision: 384
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt{e^{\log \left(1 - \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot \left(\frac{0.254829592}{\left|x\right| \cdot 0.3275911 + 1} + \frac{\frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}\right) + \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^{3}\right) \cdot \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1} + \left(\frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right)\right)\right)\right)}}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.8

    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  2. Using strategy rm

  3. Applied add-log-exp13.8

    \[\leadsto \color{blue}{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt13.8

    \[\leadsto \log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \color{blue}{\left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right)} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt13.8

    \[\leadsto \log \left(e^{\color{blue}{\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}}\right)\]

  8. Applied exp-prod13.8

    \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}^{\left(\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right)}\]

  9. Applied log-pow13.8

    \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \log \left(e^{\sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}\]

  10. Applied simplify2.1

    \[\leadsto \sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \color{blue}{\sqrt{1 - \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot \left(\frac{0.254829592}{\left|x\right| \cdot 0.3275911 + 1} + \frac{\frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}\right) + \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^{3}\right) \cdot \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1} + \left(\frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right)\right)\right)}}\]
  11. Using strategy rm

  12. Applied add-exp-log1.9

    \[\leadsto \sqrt{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \left(\left(\sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt{\color{blue}{e^{\log \left(1 - \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot \left(\frac{0.254829592}{\left|x\right| \cdot 0.3275911 + 1} + \frac{\frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}\right) + \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^{3}\right) \cdot \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1} + \left(\frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right)\right)\right)\right)}}}\]
  13. Removed slow pow expressions.

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))