logq (problem 3.4.3)

Percentage Accurate: 8.2% → 100.0%

Time: 8.6s

Alternatives: 5

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: 8.2% 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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-{\varepsilon}^{2}\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (- (log1p (- (pow eps 2.0))) (* 2.0 (log1p eps))))

C

double code(double eps) {
	return log1p(-pow(eps, 2.0)) - (2.0 * log1p(eps));
}

Java

public static double code(double eps) {
	return Math.log1p(-Math.pow(eps, 2.0)) - (2.0 * Math.log1p(eps));
}

Python

def code(eps):
	return math.log1p(-math.pow(eps, 2.0)) - (2.0 * math.log1p(eps))

Julia

function code(eps)
	return Float64(log1p(Float64(-(eps ^ 2.0))) - Float64(2.0 * log1p(eps)))
end

Wolfram

code[eps_] := N[(N[Log[1 + (-N[Power[eps, 2.0], $MachinePrecision])], $MachinePrecision] - N[(2.0 * N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.0%

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

   1. flip-- 8.9%

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

   2. associate-/l/ 9.0%

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

   3. log-div 9.0%

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

   4. metadata-eval 9.0%

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

   5. sub-neg 9.0%

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

   6. log1p-define 9.2%

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

   7. pow2 9.2%

      \[\leadsto \mathsf{log1p}\left(-\color{blue}{{\varepsilon}^{2}}\right) - \log \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \]

   8. pow2 9.2%

      \[\leadsto \mathsf{log1p}\left(-{\varepsilon}^{2}\right) - \log \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)} \]

   9. log-pow 9.2%

      \[\leadsto \mathsf{log1p}\left(-{\varepsilon}^{2}\right) - \color{blue}{2 \cdot \log \left(1 + \varepsilon\right)} \]

   10. log1p-define 100.0%

       \[\leadsto \mathsf{log1p}\left(-{\varepsilon}^{2}\right) - 2 \cdot \color{blue}{\mathsf{log1p}\left(\varepsilon\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-{\varepsilon}^{2}\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(-{\varepsilon}^{2}\right) - 2 \cdot \mathsf{log1p}\left(\varepsilon\right) \]
6. Add Preprocessing

Alternative 2: 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]

Derivation

1. Initial program 9.0%

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

   1. *-un-lft-identity 9.0%

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

   2. *-commutative 9.0%

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

   3. log-prod 9.0%

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

   4. log-div 9.1%

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

   5. sub-neg 9.1%

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

   6. log1p-define 21.5%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \left(\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\right) + \color{blue}{0} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\right) + 0} \]
5. Step-by-step derivation

   1. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right) \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left({\varepsilon}^{2} \cdot -0.6666666666666666 - 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot -0.6666666666666666 - 2\right) \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eps):
	return eps * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.0%

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

3. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 5: 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 9.0%

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

   1. *-un-lft-identity 9.0%

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

   2. *-commutative 9.0%

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

   3. log-prod 9.0%

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

   4. log-div 9.1%

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

   5. sub-neg 9.1%

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

   6. log1p-define 21.5%

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

   7. log1p-define 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \left(\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\right) + \color{blue}{0} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\right) + 0} \]
5. Step-by-step derivation

   1. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

   1. log1p-undefine 21.5%

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

   2. log1p-undefine 9.1%

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

   3. diff-log 9.0%

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

   4. add-sqr-sqrt 4.1%

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

   5. sqrt-unprod 6.5%

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

   6. sqr-neg 6.5%

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

   7. sqrt-unprod 2.5%

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

   8. add-sqr-sqrt 5.4%

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

   9. pow1 5.4%

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

   10. rem-exp-log 5.4%

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

   11. log1p-undefine 5.4%

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

   12. pow1 5.4%

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

   13. rem-exp-log 5.4%

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

   14. log1p-undefine 5.4%

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

   15. pow-div 5.4%

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

   16. metadata-eval 5.4%

       \[\leadsto \log \left({\left(e^{\mathsf{log1p}\left(\varepsilon\right)}\right)}^{\color{blue}{0}}\right) \]

   17. metadata-eval 5.4%

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

   18. metadata-eval 5.4%

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

8. Applied egg-rr 5.4%

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

9. Final simplification 5.4%

   \[\leadsto 0 \]
10. 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 2024053 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64
  :pre (< (fabs eps) 1.0)

  :alt
  (- (log1p (- eps)) (log1p eps))

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