logq (problem 3.4.3)

Percentage Accurate: 76.5% → 97.9%

Time: 6.8s

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: 76.5% 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps_s \cdot \begin{array}{l} \mathbf{if}\;eps_m \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-eps_m\right) - \mathsf{log1p}\left(eps_m\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (if (<= eps_m 2.1e-84) 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 <= 2.1e-84) {
		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 <= 2.1e-84) {
		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 <= 2.1e-84:
		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 <= 2.1e-84)
		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, 2.1e-84], 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 < 2.09999999999999998e-84

   1. Initial program 86.7%

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

   3. Taylor expanded in eps around 0 84.9%

      \[\leadsto \log \color{blue}{1} \]

   if 2.09999999999999998e-84 < eps

   1. Initial program 16.8%

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

      1. log-div 16.8%

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

      2. sub-neg 16.8%

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

      3. log1p-def 29.2%

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

      4. log1p-def 98.0%

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

   3. Simplified 98.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps_s \cdot \begin{array}{l} \mathbf{if}\;eps_m \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps_m \cdot -2 + -0.6666666666666666 \cdot {eps_m}^{3}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (*
  eps_s
  (if (<= eps_m 2.1e-84)
    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 <= 2.1e-84) {
		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 <= 2.1d-84) 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 <= 2.1e-84) {
		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 <= 2.1e-84:
		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 <= 2.1e-84)
		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 <= 2.1e-84)
		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, 2.1e-84], 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 < 2.09999999999999998e-84

   1. Initial program 86.7%

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

   3. Taylor expanded in eps around 0 84.9%

      \[\leadsto \log \color{blue}{1} \]

   if 2.09999999999999998e-84 < eps

   1. Initial program 16.8%

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

   3. Taylor expanded in eps around 0 94.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -2 + -0.6666666666666666 \cdot {\varepsilon}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps_s \cdot \begin{array}{l} \mathbf{if}\;eps_m \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps_m \cdot -2\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (if (<= eps_m 2.1e-84) 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 <= 2.1e-84) {
		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 <= 2.1d-84) 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 <= 2.1e-84) {
		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 <= 2.1e-84:
		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 <= 2.1e-84)
		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 <= 2.1e-84)
		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, 2.1e-84], 0.0, N[(eps$95$m * -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if eps < 2.09999999999999998e-84

   1. Initial program 86.7%

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

   3. Taylor expanded in eps around 0 84.9%

      \[\leadsto \log \color{blue}{1} \]

   if 2.09999999999999998e-84 < eps

   1. Initial program 16.8%

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

   3. Taylor expanded in eps around 0 92.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 28.8% accurate, 35.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps_s \cdot \left(eps_m \cdot -2\right) \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m) :precision binary64 (* eps_s (* eps_m -2.0)))

C

eps_m = fabs(eps);
eps_s = copysign(1.0, eps);
double code(double eps_s, double eps_m) {
	return eps_s * (eps_m * -2.0);
}

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 * (eps_m * (-2.0d0))
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 * (eps_m * -2.0);
}

Python

eps_m = math.fabs(eps)
eps_s = math.copysign(1.0, eps)
def code(eps_s, eps_m):
	return eps_s * (eps_m * -2.0)

Julia

eps_m = abs(eps)
eps_s = copysign(1.0, eps)
function code(eps_s, eps_m)
	return Float64(eps_s * Float64(eps_m * -2.0))
end

MATLAB

eps_m = abs(eps);
eps_s = sign(eps) * abs(1.0);
function tmp = code(eps_s, eps_m)
	tmp = eps_s * (eps_m * -2.0);
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 * N[(eps$95$m * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in eps around 0 31.2%

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

4. Final simplification 31.2%

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

Developer target: 29.7% 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 2024018 
(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))))