Average Error: 0.2 → 0.1
Time: 1.9m
Precision: 64
Internal Precision: 384
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{1}{5}\right) \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + \left((\frac{2}{3} \cdot \left({\left(\left|x\right|\right)}^{3}\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\frac{\frac{1}{21}}{\sqrt{\pi}} \cdot {\left(\left|x\right|\right)}^{7}\right))_*\right|\]

Error

Bits error versus x

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}}\right)\right|\]

  3. Taylor expanded around 0 0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right|\]

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{1}{5}\right) \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^{3}\right) + \left((\frac{2}{3} \cdot \left({\left(\left|x\right|\right)}^{3}\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left({\left(\left|x\right|\right)}^{7} \cdot \frac{\frac{1}{21} \cdot 1}{\sqrt{\pi}}\right))_*\right|}\]

  5. Applied simplify0.1

    \[\leadsto \left|\color{blue}{(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{1}{5}\right) \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + \left((\frac{2}{3} \cdot \left({\left(\left|x\right|\right)}^{3}\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\frac{\frac{1}{21}}{\sqrt{\pi}} \cdot {\left(\left|x\right|\right)}^{7}\right))_*}\right|\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))