logq (problem 3.4.3)

Average Error: 58.5 → 0.2

Time: 7.8s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((-2.0d0) * eps) + (((-0.4d0) * (eps ** 5.0d0)) + ((-0.6666666666666666d0) * (eps ** 3.0d0)))
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 (-2.0 * eps) + ((-0.4 * Math.pow(eps, 5.0)) + (-0.6666666666666666 * Math.pow(eps, 3.0)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	58.5
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 58.5

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
   Proof
   (-.f64 (log1p.f64 (neg.f64 eps)) (log1p.f64 eps)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 eps)))) (log1p.f64 eps)): 256 points increase in error, 0 points decrease in error
   (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 eps))) (log1p.f64 eps)): 0 points increase in error, 0 points decrease in error
   (-.f64 (log.f64 (-.f64 1 eps)) (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 eps)))): 249 points increase in error, 7 points decrease in error
   (Rewrite<= log-div_binary64 (log.f64 (/.f64 (-.f64 1 eps) (+.f64 1 eps)))): 11 points increase in error, 3 points decrease in error

3. Taylor expanded in eps around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.0
Cost	13056

\[\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \]

Alternative 2
Error	0.3
Cost	704

\[-2 \cdot \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right) \]

Alternative 3
Error	0.7
Cost	192

\[-2 \cdot \varepsilon \]

Reproduce

herbie shell --seed 2022317 
(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))))