expm1 (example 3.7)

?

Percentage Accurate: 40.2% → 100.0%
Time: 1.2s
Precision: binary64
Cost: 6464

?

\[-0.00017 < x\]
\[e^{x} - 1 \]
↓
\[\mathsf{expm1}\left(x\right) \]
(FPCore (x) :precision binary64 (- (exp x) 1.0))
↓
(FPCore (x) :precision binary64 (expm1 x))
double code(double x) {
	return exp(x) - 1.0;
}
↓
double code(double x) {
	return expm1(x);
}

public static double code(double x) {
	return Math.exp(x) - 1.0;
}
↓
public static double code(double x) {
	return Math.expm1(x);
}

def code(x):
	return math.exp(x) - 1.0
↓
def code(x):
	return math.expm1(x)

function code(x)
	return Float64(exp(x) - 1.0)
end
↓
function code(x)
	return expm1(x)
end

code[x_] := N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[x_] := N[(Exp[x] - 1), $MachinePrecision]

e^{x} - 1
↓
\mathsf{expm1}\left(x\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2%
Target88.2%
Herbie100.0%
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right) \]

Derivation?

  1. Initial program 39.0%

    \[e^{x} - 1 \]

  2. Simplified100.0%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
    Step-by-step derivation

    [Start]39.0

    \[ e^{x} - 1 \]

    expm1-def [=>]100.0

    \[ \color{blue}{\mathsf{expm1}\left(x\right)} \]

  3. Final simplification100.0%

    \[\leadsto \mathsf{expm1}\left(x\right) \]

Reproduce?

herbie shell --seed 2023160 
(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))