logq (problem 3.4.3)

Percentage Accurate: 8.4% → 100.0%

Time: 8.9s

Alternatives: 6

Speedup: 53.5×

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: 8.4% 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, 0.5× 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 8.3%

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

   1. *-un-lft-identity 8.3%

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

   2. *-commutative 8.3%

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

   3. log-prod 8.3%

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

   4. log-div 8.4%

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

   5. sub-neg 8.4%

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

   6. log1p-define 20.6%

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

   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. Add Preprocessing

Alternative 2: 99.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double eps) {
	return (eps * (-0.6666666666666666 * (eps * eps))) + (eps * -2.0);
}

Python

def code(eps):
	return (eps * (-0.6666666666666666 * (eps * eps))) + (eps * -2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in eps around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right)} \]
4. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. metadata-eval 99.5%

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

   3. distribute-rgt-in 99.5%

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

   4. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left(-0.6666666666666666 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + \varepsilon \cdot -2} \]
6. Step-by-step derivation

   1. unpow2 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(-0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot -2 \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double eps) {
	return eps * ((-0.6666666666666666 * (eps * eps)) - 2.0);
}

Python

def code(eps):
	return eps * ((-0.6666666666666666 * (eps * eps)) - 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.3%

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

3. Taylor expanded in eps around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(-0.6666666666666666 \cdot {\varepsilon}^{2} - 2\right)} \]
4. Step-by-step derivation

   1. unpow2 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \left(-0.6666666666666666 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 2\right) \]
6. Add Preprocessing

Alternative 4: 98.9% 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 8.3%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 18.8% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eps):
	return -eps

Julia

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

MATLAB

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

Wolfram

code[eps_] := (-eps)

Derivation

1. Initial program 8.3%

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

   1. *-un-lft-identity 8.3%

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

   2. *-commutative 8.3%

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

   3. log-prod 8.3%

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

   4. log-div 8.4%

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

   5. sub-neg 8.4%

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

   6. log1p-define 20.6%

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

   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. Taylor expanded in eps around 0 98.5%

   \[\leadsto \color{blue}{-1 \cdot \varepsilon} - \mathsf{log1p}\left(\varepsilon\right) \]
8. Step-by-step derivation

   1. mul-1-neg 98.5%

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

9. Simplified 98.5%

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

10. Taylor expanded in eps around inf 18.7%

    \[\leadsto \color{blue}{-1 \cdot \varepsilon} \]
11. Step-by-step derivation

    1. mul-1-neg 18.7%

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

12. Simplified 18.7%

    \[\leadsto \color{blue}{-\varepsilon} \]
13. Add Preprocessing

Alternative 6: 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 8.3%

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

   1. *-un-lft-identity 8.3%

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

   2. *-commutative 8.3%

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

   3. log-prod 8.3%

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

   4. log-div 8.4%

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

   5. sub-neg 8.4%

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

   6. log1p-define 20.6%

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

   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. sub-neg 100.0%

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

   2. add-sqr-sqrt 46.7%

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

   3. sqrt-unprod 31.6%

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

   4. sqr-neg 31.6%

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

   5. sqrt-prod 3.5%

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

   6. add-sqr-sqrt 5.7%

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

8. Applied egg-rr 5.7%

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

   1. sub-neg 5.7%

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

   2. +-inverses 5.7%

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

10. Simplified 5.7%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 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 2024180 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64
  :pre (< (fabs eps) 1.0)

  :alt
  (! :herbie-platform default (- (log1p (- eps)) (log1p eps)))

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