Average Error: 28.9 → 0.5
Time: 48.5s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot {\left(e^{a \cdot x} - 1\right)}^{\frac{1}{3}} \le 0.0001762820483530575:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original28.9
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (- (exp (* a x)) 1)) (cbrt (- (exp (* a x)) 1))) (pow (- (exp (* a x)) 1) 1/3)) < 0.0001762820483530575

    1. Initial program 44.5

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

    2. Taylor expanded around 0 13.0

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

    3. Applied simplify0.0

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

    if 0.0001762820483530575 < (* (* (cbrt (- (exp (* a x)) 1)) (cbrt (- (exp (* a x)) 1))) (pow (- (exp (* a x)) 1) 1/3))

    1. Initial program 1.2

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

    3. Applied add-sqr-sqrt1.3

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

    4. Applied difference-of-sqr-11.3

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

    6. Applied add-cube-cbrt1.3

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{e^{a \cdot x}} - 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right) \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 48.5s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(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))