Average Error: 58.6 → 0.4
Time: 2.7s
Precision: binary64
\[-0.00017 < x\]
\[e^{x} - 1\]
↓
\[\left(1 + \sqrt{e^{x}}\right) \cdot \left(x \cdot 0.5 + \left(0.125 \cdot {x}^{2} + \left(0.020833333333333332 \cdot {x}^{3} + 0.0026041666666666665 \cdot {x}^{4}\right)\right)\right)\]
e^{x} - 1
↓
\left(1 + \sqrt{e^{x}}\right) \cdot \left(x \cdot 0.5 + \left(0.125 \cdot {x}^{2} + \left(0.020833333333333332 \cdot {x}^{3} + 0.0026041666666666665 \cdot {x}^{4}\right)\right)\right)
(FPCore (x) :precision binary64 (- (exp x) 1.0))
↓
(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (sqrt (exp x)))
  (+
   (* x 0.5)
   (+
    (* 0.125 (pow x 2.0))
    (+
     (* 0.020833333333333332 (pow x 3.0))
     (* 0.0026041666666666665 (pow x 4.0)))))))
double code(double x) {
	return exp(x) - 1.0;
}
↓
double code(double x) {
	return (1.0 + sqrt(exp(x))) * ((x * 0.5) + ((0.125 * pow(x, 2.0)) + ((0.020833333333333332 * pow(x, 3.0)) + (0.0026041666666666665 * pow(x, 4.0)))));
}

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. Using strategy rm

  3. Applied add-sqr-sqrt_binary6458.7

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

  4. Applied difference-of-sqr-1_binary6458.7

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

  5. Simplified58.7

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

  6. Simplified58.7

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

  7. Taylor expanded around 0 0.4

    \[\leadsto \left(1 + \sqrt{e^{x}}\right) \cdot \color{blue}{\left(0.5 \cdot x + \left(0.125 \cdot {x}^{2} + \left(0.020833333333333332 \cdot {x}^{3} + 0.0026041666666666665 \cdot {x}^{4}\right)\right)\right)}\]

  8. Final simplification0.4

    \[\leadsto \left(1 + \sqrt{e^{x}}\right) \cdot \left(x \cdot 0.5 + \left(0.125 \cdot {x}^{2} + \left(0.020833333333333332 \cdot {x}^{3} + 0.0026041666666666665 \cdot {x}^{4}\right)\right)\right)\]

Reproduce

herbie shell --seed 2021128 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))