logq (problem 3.4.3)

Percentage Accurate: 8.3% → 100.0%

Time: 8.8s

Alternatives: 6

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: 8.3% 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: 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 8.6%

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

   1. *-un-lft-identity 8.6%

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

   2. *-commutative 8.6%

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

   3. log-prod 8.6%

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

   4. log-div 8.7%

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

   5. sub-neg 8.7%

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

   6. log1p-define 21.2%

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

   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: 99.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.4 - 0.6666666666666666\right) - 2\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (* eps (- (* (* eps eps) (- (* (* eps eps) -0.4) 0.6666666666666666)) 2.0)))

C

double code(double eps) {
	return eps * (((eps * eps) * (((eps * eps) * -0.4) - 0.6666666666666666)) - 2.0);
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = eps * (((eps * eps) * (((eps * eps) * (-0.4d0)) - 0.6666666666666666d0)) - 2.0d0)
end function

Java

public static double code(double eps) {
	return eps * (((eps * eps) * (((eps * eps) * -0.4) - 0.6666666666666666)) - 2.0);
}

Python

def code(eps):
	return eps * (((eps * eps) * (((eps * eps) * -0.4) - 0.6666666666666666)) - 2.0)

Julia

function code(eps)
	return Float64(eps * Float64(Float64(Float64(eps * eps) * Float64(Float64(Float64(eps * eps) * -0.4) - 0.6666666666666666)) - 2.0))
end

MATLAB

function tmp = code(eps)
	tmp = eps * (((eps * eps) * (((eps * eps) * -0.4) - 0.6666666666666666)) - 2.0);
end

Wolfram

code[eps_] := N[(eps * N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * -0.4), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in eps around 0 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot {\varepsilon}^{2} - 0.6666666666666666\right) - 2\right)} \]
4. Step-by-step derivation

   1. unpow2 99.9%

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 0.6666666666666666\right) - 2\right) \]

5. Applied egg-rr 99.9%

   \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 0.6666666666666666\right) - 2\right) \]
6. Step-by-step derivation

   1. unpow2 99.9%

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 0.6666666666666666\right) - 2\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(-0.4 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.6666666666666666\right) - 2\right) \]

8. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.4 - 0.6666666666666666\right) - 2\right) \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. sub-neg 99.7%

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

   2. metadata-eval 99.7%

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

   3. distribute-rgt-in 99.7%

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

   4. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. unpow2 99.9%

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 0.6666666666666666\right) - 2\right) \]

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. unpow2 99.9%

      \[\leadsto \varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(-0.4 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - 0.6666666666666666\right) - 2\right) \]

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.0% 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 8.6%

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

3. Taylor expanded in eps around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot -2 \]
5. 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 8.6%

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

   1. *-un-lft-identity 8.6%

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

   2. *-commutative 8.6%

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

   3. log-prod 8.6%

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

   4. log-div 8.7%

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

   5. sub-neg 8.7%

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

   6. log1p-define 21.2%

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

   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 49.8%

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

   3. sqrt-unprod 35.9%

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

   4. sqr-neg 35.9%

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

   5. sqrt-prod 3.3%

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

   6. add-sqr-sqrt 5.4%

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

8. Applied egg-rr 5.4%

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

   1. sub-neg 5.4%

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

   2. +-inverses 5.4%

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

10. Simplified 5.4%

    \[\leadsto \color{blue}{0} \]
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 2024163 
(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))))