logq (problem 3.4.3)

Percentage Accurate: 82.5% → 99.9%

Time: 5.8s

Alternatives: 4

Speedup: 13.3×

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: 82.5% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps\_s \cdot \begin{array}{l} \mathbf{if}\;eps\_m \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-eps\_m\right) - \mathsf{log1p}\left(eps\_m\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (if (<= eps_m 3.8e-68) 0.0 (- (log1p (- eps_m)) (log1p eps_m)))))

C

eps_m = fabs(eps);
eps_s = copysign(1.0, eps);
double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = log1p(-eps_m) - log1p(eps_m);
	}
	return eps_s * tmp;
}

Java

eps_m = Math.abs(eps);
eps_s = Math.copySign(1.0, eps);
public static double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = Math.log1p(-eps_m) - Math.log1p(eps_m);
	}
	return eps_s * tmp;
}

Python

eps_m = math.fabs(eps)
eps_s = math.copysign(1.0, eps)
def code(eps_s, eps_m):
	tmp = 0
	if eps_m <= 3.8e-68:
		tmp = 0.0
	else:
		tmp = math.log1p(-eps_m) - math.log1p(eps_m)
	return eps_s * tmp

Julia

eps_m = abs(eps)
eps_s = copysign(1.0, eps)
function code(eps_s, eps_m)
	tmp = 0.0
	if (eps_m <= 3.8e-68)
		tmp = 0.0;
	else
		tmp = Float64(log1p(Float64(-eps_m)) - log1p(eps_m));
	end
	return Float64(eps_s * tmp)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
eps_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[eps]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[eps$95$s_, eps$95$m_] := N[(eps$95$s * If[LessEqual[eps$95$m, 3.8e-68], 0.0, N[(N[Log[1 + (-eps$95$m)], $MachinePrecision] - N[Log[1 + eps$95$m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if eps < 3.80000000000000038e-68

   1. Initial program 90.8%

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

      1. div-inv 90.8%

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

      2. *-un-lft-identity 90.8%

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

      3. sub-neg 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. add-sqr-sqrt 43.6%

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

      6. sqrt-unprod 90.8%

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

      7. sqr-neg 90.8%

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

      8. sqrt-unprod 47.2%

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

      9. add-sqr-sqrt 88.7%

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

      10. sum-log 88.8%

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

      11. log1p-udef 5.8%

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

      12. neg-log 5.7%

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

      13. log1p-udef 88.7%

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

   4. Applied egg-rr 88.7%

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

      1. sub-neg 88.7%

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

      2. +-inverses 88.7%

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

   6. Simplified 88.7%

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

   if 3.80000000000000038e-68 < eps

   1. Initial program 15.7%

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

      1. log-div 15.7%

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

      2. sub-neg 15.7%

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

      3. log1p-def 27.3%

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

      4. log1p-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps\_s \cdot \begin{array}{l} \mathbf{if}\;eps\_m \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot -2 + -0.6666666666666666 \cdot {eps\_m}^{3}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (*
  eps_s
  (if (<= eps_m 3.8e-68)
    0.0
    (+ (* eps_m -2.0) (* -0.6666666666666666 (pow eps_m 3.0))))))

C

eps_m = fabs(eps);
eps_s = copysign(1.0, eps);
double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * -2.0) + (-0.6666666666666666 * pow(eps_m, 3.0));
	}
	return eps_s * tmp;
}

Fortran

eps_m = abs(eps)
eps_s = copysign(1.0d0, eps)
real(8) function code(eps_s, eps_m)
    real(8), intent (in) :: eps_s
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 3.8d-68) then
        tmp = 0.0d0
    else
        tmp = (eps_m * (-2.0d0)) + ((-0.6666666666666666d0) * (eps_m ** 3.0d0))
    end if
    code = eps_s * tmp
end function

Java

eps_m = Math.abs(eps);
eps_s = Math.copySign(1.0, eps);
public static double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * -2.0) + (-0.6666666666666666 * Math.pow(eps_m, 3.0));
	}
	return eps_s * tmp;
}

Python

eps_m = math.fabs(eps)
eps_s = math.copysign(1.0, eps)
def code(eps_s, eps_m):
	tmp = 0
	if eps_m <= 3.8e-68:
		tmp = 0.0
	else:
		tmp = (eps_m * -2.0) + (-0.6666666666666666 * math.pow(eps_m, 3.0))
	return eps_s * tmp

Julia

eps_m = abs(eps)
eps_s = copysign(1.0, eps)
function code(eps_s, eps_m)
	tmp = 0.0
	if (eps_m <= 3.8e-68)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * -2.0) + Float64(-0.6666666666666666 * (eps_m ^ 3.0)));
	end
	return Float64(eps_s * tmp)
end

MATLAB

eps_m = abs(eps);
eps_s = sign(eps) * abs(1.0);
function tmp_2 = code(eps_s, eps_m)
	tmp = 0.0;
	if (eps_m <= 3.8e-68)
		tmp = 0.0;
	else
		tmp = (eps_m * -2.0) + (-0.6666666666666666 * (eps_m ^ 3.0));
	end
	tmp_2 = eps_s * tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
eps_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[eps]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[eps$95$s_, eps$95$m_] := N[(eps$95$s * If[LessEqual[eps$95$m, 3.8e-68], 0.0, N[(N[(eps$95$m * -2.0), $MachinePrecision] + N[(-0.6666666666666666 * N[Power[eps$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if eps < 3.80000000000000038e-68

   1. Initial program 90.8%

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

      1. div-inv 90.8%

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

      2. *-un-lft-identity 90.8%

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

      3. sub-neg 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. add-sqr-sqrt 43.6%

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

      6. sqrt-unprod 90.8%

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

      7. sqr-neg 90.8%

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

      8. sqrt-unprod 47.2%

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

      9. add-sqr-sqrt 88.7%

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

      10. sum-log 88.8%

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

      11. log1p-udef 5.8%

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

      12. neg-log 5.7%

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

      13. log1p-udef 88.7%

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

   4. Applied egg-rr 88.7%

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

      1. sub-neg 88.7%

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

      2. +-inverses 88.7%

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

   6. Simplified 88.7%

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

   if 3.80000000000000038e-68 < eps

   1. Initial program 15.7%

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

   3. Taylor expanded in eps around 0 98.2%

      \[\leadsto \color{blue}{-2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -2 + -0.6666666666666666 \cdot {\varepsilon}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps\_s \cdot \begin{array}{l} \mathbf{if}\;eps\_m \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot -2\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (if (<= eps_m 3.8e-68) 0.0 (* eps_m -2.0))))

C

eps_m = fabs(eps);
eps_s = copysign(1.0, eps);
double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = eps_m * -2.0;
	}
	return eps_s * tmp;
}

Fortran

eps_m = abs(eps)
eps_s = copysign(1.0d0, eps)
real(8) function code(eps_s, eps_m)
    real(8), intent (in) :: eps_s
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 3.8d-68) then
        tmp = 0.0d0
    else
        tmp = eps_m * (-2.0d0)
    end if
    code = eps_s * tmp
end function

Java

eps_m = Math.abs(eps);
eps_s = Math.copySign(1.0, eps);
public static double code(double eps_s, double eps_m) {
	double tmp;
	if (eps_m <= 3.8e-68) {
		tmp = 0.0;
	} else {
		tmp = eps_m * -2.0;
	}
	return eps_s * tmp;
}

Python

eps_m = math.fabs(eps)
eps_s = math.copysign(1.0, eps)
def code(eps_s, eps_m):
	tmp = 0
	if eps_m <= 3.8e-68:
		tmp = 0.0
	else:
		tmp = eps_m * -2.0
	return eps_s * tmp

Julia

eps_m = abs(eps)
eps_s = copysign(1.0, eps)
function code(eps_s, eps_m)
	tmp = 0.0
	if (eps_m <= 3.8e-68)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * -2.0);
	end
	return Float64(eps_s * tmp)
end

MATLAB

eps_m = abs(eps);
eps_s = sign(eps) * abs(1.0);
function tmp_2 = code(eps_s, eps_m)
	tmp = 0.0;
	if (eps_m <= 3.8e-68)
		tmp = 0.0;
	else
		tmp = eps_m * -2.0;
	end
	tmp_2 = eps_s * tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
eps_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[eps]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[eps$95$s_, eps$95$m_] := N[(eps$95$s * If[LessEqual[eps$95$m, 3.8e-68], 0.0, N[(eps$95$m * -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if eps < 3.80000000000000038e-68

   1. Initial program 90.8%

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

      1. div-inv 90.8%

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

      2. *-un-lft-identity 90.8%

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

      3. sub-neg 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. add-sqr-sqrt 43.6%

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

      6. sqrt-unprod 90.8%

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

      7. sqr-neg 90.8%

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

      8. sqrt-unprod 47.2%

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

      9. add-sqr-sqrt 88.7%

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

      10. sum-log 88.8%

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

      11. log1p-udef 5.8%

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

      12. neg-log 5.7%

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

      13. log1p-udef 88.7%

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

   4. Applied egg-rr 88.7%

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

      1. sub-neg 88.7%

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

      2. +-inverses 88.7%

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

   6. Simplified 88.7%

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

   if 3.80000000000000038e-68 < eps

   1. Initial program 15.7%

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

   3. Taylor expanded in eps around 0 97.0%

      \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.8 \cdot 10^{-68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps\_s \cdot 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m) :precision binary64 (* eps_s 0.0))

C

eps_m = fabs(eps);
eps_s = copysign(1.0, eps);
double code(double eps_s, double eps_m) {
	return eps_s * 0.0;
}

Fortran

eps_m = abs(eps)
eps_s = copysign(1.0d0, eps)
real(8) function code(eps_s, eps_m)
    real(8), intent (in) :: eps_s
    real(8), intent (in) :: eps_m
    code = eps_s * 0.0d0
end function

Java

eps_m = Math.abs(eps);
eps_s = Math.copySign(1.0, eps);
public static double code(double eps_s, double eps_m) {
	return eps_s * 0.0;
}

Python

eps_m = math.fabs(eps)
eps_s = math.copysign(1.0, eps)
def code(eps_s, eps_m):
	return eps_s * 0.0

Julia

eps_m = abs(eps)
eps_s = copysign(1.0, eps)
function code(eps_s, eps_m)
	return Float64(eps_s * 0.0)
end

MATLAB

eps_m = abs(eps);
eps_s = sign(eps) * abs(1.0);
function tmp = code(eps_s, eps_m)
	tmp = eps_s * 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
eps_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[eps]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[eps$95$s_, eps$95$m_] := N[(eps$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. div-inv 83.4%

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

   2. *-un-lft-identity 83.4%

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

   3. sub-neg 83.4%

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

   4. *-un-lft-identity 83.4%

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

   5. add-sqr-sqrt 39.3%

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

   6. sqrt-unprod 82.2%

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

   7. sqr-neg 82.2%

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

   8. sqrt-unprod 42.9%

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

   9. add-sqr-sqrt 80.4%

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

   10. sum-log 80.7%

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

   11. log1p-udef 5.5%

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

   12. neg-log 5.4%

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

   13. log1p-udef 80.4%

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

4. Applied egg-rr 80.4%

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

   1. sub-neg 80.4%

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

   2. +-inverses 80.4%

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

6. Simplified 80.4%

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

7. Final simplification 80.4%

   \[\leadsto 0 \]
8. Add Preprocessing

Developer target: 26.0% 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 2024031 
(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))))