Average Error: 58.6 → 0.5
Time: 11.4s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[\left(1 + \sqrt{e^{x}}\right) \cdot \left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\frac{1}{2} + \frac{1}{48} \cdot \left(x \cdot x\right)\right) \cdot x\right)\]
e^{x} - 1
↓
\left(1 + \sqrt{e^{x}}\right) \cdot \left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\frac{1}{2} + \frac{1}{48} \cdot \left(x \cdot x\right)\right) \cdot x\right)
double f(double x) {
        double r2293229 = x;
        double r2293230 = exp(r2293229);
        double r2293231 = 1.0;
        double r2293232 = r2293230 - r2293231;
        return r2293232;
}
↓
double f(double x) {
        double r2293233 = 1.0;
        double r2293234 = x;
        double r2293235 = exp(r2293234);
        double r2293236 = sqrt(r2293235);
        double r2293237 = r2293233 + r2293236;
        double r2293238 = 0.125;
        double r2293239 = r2293234 * r2293234;
        double r2293240 = r2293238 * r2293239;
        double r2293241 = 0.5;
        double r2293242 = 0.020833333333333332;
        double r2293243 = r2293242 * r2293239;
        double r2293244 = r2293241 + r2293243;
        double r2293245 = r2293244 * r2293234;
        double r2293246 = r2293240 + r2293245;
        double r2293247 = r2293237 * r2293246;
        return r2293247;
}

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 add-sqr-sqrt58.6

    \[\leadsto \color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - 1\]

  4. Applied difference-of-sqr-158.6

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

  5. Taylor expanded around 0 0.5

    \[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{8} \cdot {x}^{2}\right)\right)}\]

  6. Simplified0.5

    \[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \frac{1}{48}\right) + \left(x \cdot x\right) \cdot \frac{1}{8}\right)}\]

  7. Final simplification0.5

    \[\leadsto \left(1 + \sqrt{e^{x}}\right) \cdot \left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\frac{1}{2} + \frac{1}{48} \cdot \left(x \cdot x\right)\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))