Average Error: 58.6 → 0.4
Time: 11.7s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\left(\frac{{x}^{2}}{2} + x\right) + \frac{{x}^{3}}{6}\]
e^{x} - 1
↓
\left(\frac{{x}^{2}}{2} + x\right) + \frac{{x}^{3}}{6}
double f(double x) {
        double r70041 = x;
        double r70042 = exp(r70041);
        double r70043 = 1.0;
        double r70044 = r70042 - r70043;
        return r70044;
}
↓
double f(double x) {
        double r70045 = x;
        double r70046 = 2.0;
        double r70047 = pow(r70045, r70046);
        double r70048 = r70047 / r70046;
        double r70049 = r70048 + r70045;
        double r70050 = 3.0;
        double r70051 = pow(r70045, r70050);
        double r70052 = 6.0;
        double r70053 = r70051 / r70052;
        double r70054 = r70049 + r70053;
        return r70054;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.5
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto \left(\frac{{x}^{2}}{2} + x\right) + \frac{{x}^{3}}{6}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -1.7e-4 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))