expax (section 3.5)

Average Error: 29.2 → 9.2

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{{\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + \left(-{1}^{6}\right)\right)}^{3}}{{\left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

C

double code(double a, double x) {
	return (exp((a * x)) - 1.0);
}

↓

double code(double a, double x) {
	double temp;
	if (((a * x) <= -3.569795479894664e-07)) {
		temp = (cbrt((pow((exp((((a * x) * 3.0) + ((a * x) * 3.0))) + -pow(1.0, 6.0)), 3.0) / pow((exp(((a * x) * 3.0)) + pow(1.0, 3.0)), 3.0))) / ((exp((a * x)) * (exp((a * x)) + 1.0)) + (1.0 * 1.0)));
	} else {
		double temp_1;
		if (((a * x) <= 2.9747332644534994e-19)) {
			temp_1 = ((x * (a + ((0.5 * pow(a, 2.0)) * x))) + (0.16666666666666666 * (pow(a, 3.0) * pow(x, 3.0))));
		} else {
			temp_1 = (cbrt(pow((exp(((a * x) * 3.0)) - pow(1.0, 3.0)), 3.0)) / ((exp((a * x)) * (exp((a * x)) + 1.0)) + (1.0 * 1.0)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.2
Target	0.2
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if (* a x) < -3.569795479894664e-07

   1. Initial program 0.2

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

   3. Applied flip3-- 0.2

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

   4. Simplified 0.2

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

   6. Applied pow-exp 0.2

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

   8. Applied add-cbrt-cube 0.2

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

   9. Simplified 0.2

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

   11. Applied flip-- 0.2

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

   12. Applied cube-div 0.2

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

   13. Simplified 0.2

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

   if -3.569795479894664e-07 < (* a x) < 2.9747332644534994e-19

   1. Initial program 45.1

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

   2. Taylor expanded around 0 13.5

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

   3. Simplified 13.5

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

   if 2.9747332644534994e-19 < (* a x)

   1. Initial program 25.4

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

   3. Applied flip3-- 26.4

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

   4. Simplified 26.4

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

   6. Applied pow-exp 25.0

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

   8. Applied add-cbrt-cube 26.9

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

   9. Simplified 26.9

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{{\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + \left(-{1}^{6}\right)\right)}^{3}}{{\left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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