expq2 (section 3.11)

Average Error: 41.0 → 1.6

Time: 2.8s

Precision: binary64

Cost: 6592

Math

\[\frac{e^{x}}{e^{x} - 1} \]

↓

\[\frac{e^{x}}{x} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))

↓

(FPCore (x) :precision binary64 (/ (exp x) x))

C

double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}

↓

double code(double x) {
	return exp(x) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) / (exp(x) - 1.0d0)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) / x
end function

Java

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

↓

public static double code(double x) {
	return Math.exp(x) / x;
}

Python

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

↓

def code(x):
	return math.exp(x) / x

Julia

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end

↓

function code(x)
	return Float64(exp(x) / x)
end

MATLAB

function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end

↓

function tmp = code(x)
	tmp = exp(x) / x;
end

Wolfram

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

↓

code[x_] := N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision]

Target

Original	41.0
Target	40.5
Herbie	1.6

\[\frac{1}{1 - e^{-x}} \]

Derivation

1. Initial program 41.0

   \[\frac{e^{x}}{e^{x} - 1} \]

2. Taylor expanded in x around 0 1.6

   \[\leadsto \frac{e^{x}}{\color{blue}{x}} \]

3. Final simplification 1.6

   \[\leadsto \frac{e^{x}}{x} \]

Alternatives

Alternative 1
Error	21.7
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))