logq (problem 3.4.3)

Percentage Accurate: 97.3% → 97.4%

Time: 7.9s

Alternatives: 3

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: 97.3% 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: 97.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ eps_s = \mathsf{copysign}\left(1, \varepsilon\right) \\ eps\_s \cdot \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot -2\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (if (<= eps_m 1.9e-20) 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 <= 1.9e-20) {
		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 <= 1.9d-20) 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 <= 1.9e-20) {
		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 <= 1.9e-20:
		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 <= 1.9e-20)
		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 <= 1.9e-20)
		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, 1.9e-20], 0.0, N[(eps$95$m * -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if eps < 1.8999999999999999e-20

   1. Initial program 98.7%

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

      1. log-div 98.7%

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

      2. sub-neg 98.7%

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

      3. log1p-define 7.8%

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

      4. log1p-define 8.3%

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

   3. Simplified 8.3%

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

      1. add-exp-log 2.6%

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

   6. Applied egg-rr 2.6%

      \[\leadsto \mathsf{log1p}\left(-\varepsilon\right) - \color{blue}{e^{\log \left(\mathsf{log1p}\left(\varepsilon\right)\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 8.3%

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

      2. sub-neg 8.3%

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

      3. neg-mul-1 8.3%

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

      4. add-sqr-sqrt 5.6%

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

      5. sqrt-unprod 33.4%

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

      6. neg-mul-1 33.4%

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

      7. neg-mul-1 33.4%

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

      8. sqr-neg 33.4%

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

      9. sqrt-unprod 26.8%

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

      10. add-sqr-sqrt 96.4%

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

   8. Applied egg-rr 96.4%

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

      1. sub-neg 96.4%

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

      2. +-inverses 96.4%

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

   10. Simplified 96.4%

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

   if 1.8999999999999999e-20 < eps

   1. Initial program 42.3%

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

   3. Taylor expanded in eps around 0 44.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
eps_s = (copysign.f64 1 eps)
(FPCore (eps_s eps_m)
 :precision binary64
 (* eps_s (log (/ (- 1.0 eps_m) (+ 1.0 eps_m)))))

C

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

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 * log(((1.0d0 - eps_m) / (1.0d0 + eps_m)))
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 * Math.log(((1.0 - eps_m) / (1.0 + eps_m)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.2%

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

3. Final simplification 96.2%

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

Alternative 3: 94.4% 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 96.2%

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

   1. log-div 96.3%

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

   2. sub-neg 96.3%

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

   3. log1p-define 9.6%

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

   4. log1p-define 10.5%

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

3. Simplified 10.5%

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

   1. add-exp-log 5.1%

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

6. Applied egg-rr 5.1%

   \[\leadsto \mathsf{log1p}\left(-\varepsilon\right) - \color{blue}{e^{\log \left(\mathsf{log1p}\left(\varepsilon\right)\right)}} \]
7. Step-by-step derivation

   1. rem-exp-log 10.5%

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

   2. sub-neg 10.5%

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

   3. neg-mul-1 10.5%

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

   4. add-sqr-sqrt 5.4%

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

   5. sqrt-unprod 32.1%

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

   6. neg-mul-1 32.1%

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

   7. neg-mul-1 32.1%

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

   8. sqr-neg 32.1%

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

   9. sqrt-unprod 25.9%

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

   10. add-sqr-sqrt 92.4%

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

8. Applied egg-rr 92.4%

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

   1. sub-neg 92.4%

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

   2. +-inverses 92.4%

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

10. Simplified 92.4%

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

11. Final simplification 92.4%

    \[\leadsto 0 \]
12. Add Preprocessing

Developer target: 9.3% 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 2024046 
(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))))