Average Error: 58.6 → 0.5
Time: 11.8s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[\left(x + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
\left(x + \frac{1}{2} \cdot \left(x \cdot x\right)\right) + \left(x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r945500 = x;
        double r945501 = exp(r945500);
        double r945502 = 1.0;
        double r945503 = r945501 - r945502;
        return r945503;
}
↓
double f(double x) {
        double r945504 = x;
        double r945505 = 0.5;
        double r945506 = r945504 * r945504;
        double r945507 = r945505 * r945506;
        double r945508 = r945504 + r945507;
        double r945509 = 0.16666666666666666;
        double r945510 = r945504 * r945509;
        double r945511 = r945510 * r945506;
        double r945512 = r945508 + r945511;
        return r945512;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  3. Applied flip3--58.6

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

  4. Simplified58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))