Average Error: 58.5 → 0.6
Time: 4.7s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x
double f(double x) {
        double r78083 = x;
        double r78084 = exp(r78083);
        double r78085 = 1.0;
        double r78086 = r78084 - r78085;
        return r78086;
}
↓
double f(double x) {
        double r78087 = x;
        double r78088 = 2.0;
        double r78089 = pow(r78087, r78088);
        double r78090 = 0.16666666666666666;
        double r78091 = r78087 * r78090;
        double r78092 = 0.5;
        double r78093 = r78091 + r78092;
        double r78094 = r78089 * r78093;
        double r78095 = r78094 + r78087;
        return r78095;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

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

  (- (exp x) 1))