expax (section 3.5)

Average Error: 29.7 → 9.5

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.010954495328794143:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 6} + \left(-{1}^{6}\right)}{{\left(e^{a \cdot x}\right)}^{3} + {1}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original	29.7
Target	0.2
Herbie	9.5

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* a x) < -0.010954495328794143

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

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

   6. Applied flip-- 0.0

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

   7. Simplified 0.0

      \[\leadsto \frac{\frac{\color{blue}{{\left(e^{a \cdot x}\right)}^{6} + \left(-{1}^{6}\right)}}{{\left(e^{a \cdot x}\right)}^{3} + {1}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\]
   8. Using strategy rm

   9. Applied pow-exp 0.0

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

   if -0.010954495328794143 < (* a x)

   1. Initial program 44.3

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

   2. Taylor expanded around 0 14.1

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

   3. Simplified 14.1

      \[\leadsto \color{blue}{x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.010954495328794143:\\ \;\;\;\;\frac{\frac{e^{\left(a \cdot x\right) \cdot 6} + \left(-{1}^{6}\right)}{{\left(e^{a \cdot x}\right)}^{3} + {1}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))