logq (problem 3.4.3)

?

Percentage Accurate: 8.7% → 99.6%
Time: 8.2s
Precision: binary64
Cost: 7424

?

\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
↓
\[-2 \cdot \varepsilon + \left(-0.4 \cdot {\varepsilon}^{5} + -0.6666666666666666 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]
(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 (* eps (* eps eps))))))
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 * (eps * (eps * eps))));
}

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 * (eps * eps))))
end function

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 * (eps * (eps * eps))));
}

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 * (eps * (eps * eps))))

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 * Float64(eps * Float64(eps * eps)))))
end

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 * (eps * eps))));
end

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[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-2 \cdot \varepsilon + \left(-0.4 \cdot {\varepsilon}^{5} + -0.6666666666666666 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)

Local Percentage Accuracy vs ?

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.

Herbie found 4 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.7%
Target99.6%
Herbie99.6%
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \]

Derivation?

  1. Initial program 7.9%

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

  2. Taylor expanded in eps around 0 100.0%

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

  3. Applied egg-rr100.0%

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

    [Start]100.0%

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

    unpow3 [=>]100.0%

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

  4. Final simplification100.0%

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

Alternatives

Alternative 1
Accuracy99.6%
Cost7424
\[-2 \cdot \varepsilon + \left(-0.4 \cdot {\varepsilon}^{5} + -0.6666666666666666 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]
Alternative 2
Accuracy99.4%
Cost6848
\[\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.6666666666666666, -2\right) \]
Alternative 3
Accuracy98.8%
Cost192
\[-2 \cdot \varepsilon \]
Alternative 4
Accuracy5.4%
Cost64
\[0 \]

Reproduce?

herbie shell --seed 2023263 
(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))))