logq (problem 3.4.3)

Average Error: 58.6 → 0.1

Time: 5.8s

Precision: binary64

Cost: 20352

Math

\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

↓

\[\left(-0.6666666666666666 \cdot {\varepsilon}^{3} + -0.2857142857142857 \cdot {\varepsilon}^{7}\right) + \left(-0.4 \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right) \]

FPCore

(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))

↓

(FPCore (eps)
 :precision binary64
 (+
  (+
   (* -0.6666666666666666 (pow eps 3.0))
   (* -0.2857142857142857 (pow eps 7.0)))
  (+ (* -0.4 (pow eps 5.0)) (* -2.0 eps))))

C

double code(double eps) {
	return log(((1.0 - eps) / (1.0 + eps)));
}

↓

double code(double eps) {
	return ((-0.6666666666666666 * pow(eps, 3.0)) + (-0.2857142857142857 * pow(eps, 7.0))) + ((-0.4 * pow(eps, 5.0)) + (-2.0 * eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = log(((1.0d0 - eps) / (1.0d0 + eps)))
end function

↓

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (((-0.6666666666666666d0) * (eps ** 3.0d0)) + ((-0.2857142857142857d0) * (eps ** 7.0d0))) + (((-0.4d0) * (eps ** 5.0d0)) + ((-2.0d0) * eps))
end function

Java

public static double code(double eps) {
	return Math.log(((1.0 - eps) / (1.0 + eps)));
}

↓

public static double code(double eps) {
	return ((-0.6666666666666666 * Math.pow(eps, 3.0)) + (-0.2857142857142857 * Math.pow(eps, 7.0))) + ((-0.4 * Math.pow(eps, 5.0)) + (-2.0 * eps));
}

Python

def code(eps):
	return math.log(((1.0 - eps) / (1.0 + eps)))

↓

def code(eps):
	return ((-0.6666666666666666 * math.pow(eps, 3.0)) + (-0.2857142857142857 * math.pow(eps, 7.0))) + ((-0.4 * math.pow(eps, 5.0)) + (-2.0 * eps))

Julia

function code(eps)
	return log(Float64(Float64(1.0 - eps) / Float64(1.0 + eps)))
end

↓

function code(eps)
	return Float64(Float64(Float64(-0.6666666666666666 * (eps ^ 3.0)) + Float64(-0.2857142857142857 * (eps ^ 7.0))) + Float64(Float64(-0.4 * (eps ^ 5.0)) + Float64(-2.0 * eps)))
end

MATLAB

function tmp = code(eps)
	tmp = log(((1.0 - eps) / (1.0 + eps)));
end

↓

function tmp = code(eps)
	tmp = ((-0.6666666666666666 * (eps ^ 3.0)) + (-0.2857142857142857 * (eps ^ 7.0))) + ((-0.4 * (eps ^ 5.0)) + (-2.0 * eps));
end

Wolfram

code[eps_] := N[Log[N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[eps_] := N[(N[(N[(-0.6666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.2857142857142857 * N[Power[eps, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.4 * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.6
Target	0.2
Herbie	0.1

\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation

1. Initial program 58.6

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]

2. Simplified 58.6

   \[\leadsto \color{blue}{\log \left(\frac{1 - \varepsilon}{\varepsilon - -1}\right)} \]
   Proof

   [Start] 58.6	\[ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
   rational_best-simplify-17 [=>] 58.6	\[ \log \left(\frac{1 - \varepsilon}{\color{blue}{\varepsilon - -1}}\right) \]

3. Taylor expanded in eps around 0 0.1

   \[\leadsto \color{blue}{-2 \cdot \varepsilon + \left(-0.4 \cdot {\varepsilon}^{5} + \left(-0.2857142857142857 \cdot {\varepsilon}^{7} + -0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{\left(-0.6666666666666666 \cdot {\varepsilon}^{3} + -0.2857142857142857 \cdot {\varepsilon}^{7}\right) + \left(-0.4 \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)} \]
   Proof

   [Start] 0.1	\[ -2 \cdot \varepsilon + \left(-0.4 \cdot {\varepsilon}^{5} + \left(-0.2857142857142857 \cdot {\varepsilon}^{7} + -0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]
   rational_best-simplify-43 [=>] 0.1	\[ \color{blue}{\left(-0.2857142857142857 \cdot {\varepsilon}^{7} + -0.6666666666666666 \cdot {\varepsilon}^{3}\right) + \left(-0.4 \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \color{blue}{\left(-0.6666666666666666 \cdot {\varepsilon}^{3} + -0.2857142857142857 \cdot {\varepsilon}^{7}\right)} + \left(-0.4 \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right) \]

5. Final simplification 0.1

   \[\leadsto \left(-0.6666666666666666 \cdot {\varepsilon}^{3} + -0.2857142857142857 \cdot {\varepsilon}^{7}\right) + \left(-0.4 \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	6912

\[-2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3} \]

Alternative 2
Error	0.6
Cost	192

\[-2 \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023097 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))