Average Error: 29.6 → 0.1
Time: 1.4m
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right) - \left(a \cdot x\right) \cdot \left(a \cdot x\right)}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) - a \cdot x} \le -0.0010555263000005984:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\right)}^{3}}\\ \mathbf{if}\;\frac{\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right) \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right)\right) - \left(a \cdot x\right) \cdot \left(a \cdot x\right)}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) - a \cdot x} \le 0.009748396594961219:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(e^{x \cdot a} - 1\right)}^{3}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2))) (* (* a x) (* a x))) (- (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* a x))) < -0.0010555263000005984

    1. Initial program 0.1

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]

    4. Applied simplify0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{x \cdot a} - 1\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.1

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{e^{x \cdot a} - 1} \cdot \sqrt[3]{e^{x \cdot a} - 1}\right) \cdot \sqrt[3]{e^{x \cdot a} - 1}\right)}}^{3}}\]

    if -0.0010555263000005984 < (/ (- (* (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2))) (* (* a x) (* a x))) (- (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* a x))) < 0.009748396594961219

    1. Initial program 57.8

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 18.2

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)}\]

    3. Applied simplify0.2

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x}\]

    if 0.009748396594961219 < (/ (- (* (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2))) (* (* a x) (* a x))) (- (* (* (* a x) (* a x)) (+ (* (* a x) 1/6) 1/2)) (* a x)))

    1. Initial program 0.0

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]

    4. Applied simplify0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{x \cdot a} - 1\right)}^{3}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))