logq (problem 3.4.3)

Percentage Accurate: 8.3% → 99.5%

Time: 8.0s

Alternatives: 3

Speedup: 35.7×

Specification

\[\left|\varepsilon\right| < 1\]

Math

\[\begin{array}{l} \\ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 8.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto -2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (* -2.0 eps))

C

double code(double eps) {
	return -2.0 * eps;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (-2.0d0) * eps
end function

Java

public static double code(double eps) {
	return -2.0 * eps;
}

Python

def code(eps):
	return -2.0 * eps

Julia

function code(eps)
	return Float64(-2.0 * eps)
end

MATLAB

function tmp = code(eps)
	tmp = -2.0 * eps;
end

Wolfram

code[eps_] := N[(-2.0 * eps), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]

4. Final simplification 99.7%

   \[\leadsto -2 \cdot \varepsilon \]
5. Add Preprocessing

Alternative 3: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 0.0)

C

double code(double eps) {
	return 0.0;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double eps) {
	return 0.0;
}

Python

def code(eps):
	return 0.0

Julia

function code(eps)
	return 0.0
end

MATLAB

function tmp = code(eps)
	tmp = 0.0;
end

Wolfram

code[eps_] := 0.0

Derivation

1. Initial program 7.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv 7.2%

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

   2. *-un-lft-identity 7.2%

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

   3. sub-neg 7.2%

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

   4. *-un-lft-identity 7.2%

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

   5. add-sqr-sqrt 3.7%

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

   6. sqrt-unprod 6.6%

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

   7. sqr-neg 6.6%

      \[\leadsto \log \left(\left(1 + \sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}\right) \cdot \frac{1}{1 + \varepsilon}\right) \]

   8. sqrt-unprod 2.9%

      \[\leadsto \log \left(\left(1 + \color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}\right) \cdot \frac{1}{1 + \varepsilon}\right) \]

   9. add-sqr-sqrt 5.5%

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

   10. sum-log 5.7%

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

   11. log1p-udef 4.2%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\varepsilon\right)} + \log \left(\frac{1}{1 + \varepsilon}\right) \]

   12. neg-log 4.1%

       \[\leadsto \mathsf{log1p}\left(\varepsilon\right) + \color{blue}{\left(-\log \left(1 + \varepsilon\right)\right)} \]

   13. log1p-udef 5.5%

       \[\leadsto \mathsf{log1p}\left(\varepsilon\right) + \left(-\color{blue}{\mathsf{log1p}\left(\varepsilon\right)}\right) \]

4. Applied egg-rr 5.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\varepsilon\right) + \left(-\mathsf{log1p}\left(\varepsilon\right)\right)} \]
5. Step-by-step derivation

   1. sub-neg 5.5%

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

   2. +-inverses 5.5%

      \[\leadsto \color{blue}{0} \]

6. Simplified 5.5%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 5.5%

   \[\leadsto 0 \]
8. Add Preprocessing

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (- (log1p (- eps)) (log1p eps)))

C

double code(double eps) {
	return log1p(-eps) - log1p(eps);
}

Java

public static double code(double eps) {
	return Math.log1p(-eps) - Math.log1p(eps);
}

Python

def code(eps):
	return math.log1p(-eps) - math.log1p(eps)

Julia

function code(eps)
	return Float64(log1p(Float64(-eps)) - log1p(eps))
end

Wolfram

code[eps_] := N[(N[Log[1 + (-eps)], $MachinePrecision] - N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024040 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64
  :pre (< (fabs eps) 1.0)

  :herbie-target
  (- (log1p (- eps)) (log1p eps))

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