logq (problem 3.4.3)

Percentage Accurate: 8.6% → 100.0%

Time: 9.3s

Alternatives: 5

Speedup: 35.7×

Specification

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: 8.6% 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 8.8%

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

   1. log-div 8.7%

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

   2. sub-neg 8.7%

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

   3. log1p-def 21.4%

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

   4. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot -2 + -0.6666666666666666 \cdot {\varepsilon}^{3} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (+ (* eps -2.0) (* -0.6666666666666666 (pow eps 3.0))))

C

double code(double eps) {
	return (eps * -2.0) + (-0.6666666666666666 * pow(eps, 3.0));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps * (-2.0d0)) + ((-0.6666666666666666d0) * (eps ** 3.0d0))
end function

Java

public static double code(double eps) {
	return (eps * -2.0) + (-0.6666666666666666 * Math.pow(eps, 3.0));
}

Python

def code(eps):
	return (eps * -2.0) + (-0.6666666666666666 * math.pow(eps, 3.0))

Julia

function code(eps)
	return Float64(Float64(eps * -2.0) + Float64(-0.6666666666666666 * (eps ^ 3.0)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps * -2.0) + (-0.6666666666666666 * (eps ^ 3.0));
end

Wolfram

code[eps_] := N[(N[(eps * -2.0), $MachinePrecision] + N[(-0.6666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

2. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{-2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3}} \]

3. Final simplification 99.4%

   \[\leadsto \varepsilon \cdot -2 + -0.6666666666666666 \cdot {\varepsilon}^{3} \]

Alternative 3: 99.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\varepsilon \cdot 0.16666666666666666 - \frac{0.5}{\varepsilon}} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/ 1.0 (- (* eps 0.16666666666666666) (/ 0.5 eps))))

C

double code(double eps) {
	return 1.0 / ((eps * 0.16666666666666666) - (0.5 / eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = 1.0d0 / ((eps * 0.16666666666666666d0) - (0.5d0 / eps))
end function

Java

public static double code(double eps) {
	return 1.0 / ((eps * 0.16666666666666666) - (0.5 / eps));
}

Python

def code(eps):
	return 1.0 / ((eps * 0.16666666666666666) - (0.5 / eps))

Julia

function code(eps)
	return Float64(1.0 / Float64(Float64(eps * 0.16666666666666666) - Float64(0.5 / eps)))
end

MATLAB

function tmp = code(eps)
	tmp = 1.0 / ((eps * 0.16666666666666666) - (0.5 / eps));
end

Wolfram

code[eps_] := N[(1.0 / N[(N[(eps * 0.16666666666666666), $MachinePrecision] - N[(0.5 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. div-sub 8.8%

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

   2. sub-neg 8.8%

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

3. Applied egg-rr 8.8%

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

   1. sub-neg 8.8%

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

   2. +-commutative 8.8%

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

   3. +-commutative 8.8%

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

5. Simplified 8.8%

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

   1. sub-div 8.8%

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

   2. clear-num 8.8%

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

   3. +-commutative 8.8%

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

7. Applied egg-rr 8.8%

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

   1. clear-num 8.8%

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

   2. diff-log 8.7%

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

   3. log1p-udef 21.4%

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

   4. flip-- 16.2%

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

   5. clear-num 16.2%

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

   6. *-un-lft-identity 16.2%

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

   7. associate-/l* 16.2%

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

   8. flip-- 21.4%

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

   9. sub-neg 21.4%

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

   10. log1p-def 99.7%

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in eps around 0 99.2%

    \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot \varepsilon - 0.5 \cdot \frac{1}{\varepsilon}}} \]
11. Step-by-step derivation

    1. *-commutative 99.2%

       \[\leadsto \frac{1}{\color{blue}{\varepsilon \cdot 0.16666666666666666} - 0.5 \cdot \frac{1}{\varepsilon}} \]

    2. associate-*r/ 99.2%

       \[\leadsto \frac{1}{\varepsilon \cdot 0.16666666666666666 - \color{blue}{\frac{0.5 \cdot 1}{\varepsilon}}} \]

    3. metadata-eval 99.2%

       \[\leadsto \frac{1}{\varepsilon \cdot 0.16666666666666666 - \frac{\color{blue}{0.5}}{\varepsilon}} \]

12. Simplified 99.2%

    \[\leadsto \frac{1}{\color{blue}{\varepsilon \cdot 0.16666666666666666 - \frac{0.5}{\varepsilon}}} \]

13. Final simplification 99.2%

    \[\leadsto \frac{1}{\varepsilon \cdot 0.16666666666666666 - \frac{0.5}{\varepsilon}} \]

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 8.8%

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

2. Taylor expanded in eps around 0 98.8%

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

3. Final simplification 98.8%

   \[\leadsto \varepsilon \cdot -2 \]

Alternative 5: 5.3% 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 8.8%

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

   1. log-div 8.7%

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

   2. sub-neg 8.7%

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

   3. log1p-udef 21.4%

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

   4. log1p-udef 100.0%

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

   5. sub-neg 100.0%

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

   6. add-sqr-sqrt 50.6%

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

   7. sqrt-unprod 36.7%

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

   8. sqr-neg 36.7%

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

   9. sqrt-unprod 3.1%

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

   10. add-sqr-sqrt 5.3%

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

3. Applied egg-rr 5.3%

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

   1. sub-neg 5.3%

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

   2. +-inverses 5.3%

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

5. Simplified 5.3%

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

6. Final simplification 5.3%

   \[\leadsto 0 \]

Developer target: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0))))

C

double code(double eps) {
	return -2.0 * ((eps + (pow(eps, 3.0) / 3.0)) + (pow(eps, 5.0) / 5.0));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (-2.0d0) * ((eps + ((eps ** 3.0d0) / 3.0d0)) + ((eps ** 5.0d0) / 5.0d0))
end function

Java

public static double code(double eps) {
	return -2.0 * ((eps + (Math.pow(eps, 3.0) / 3.0)) + (Math.pow(eps, 5.0) / 5.0));
}

Python

def code(eps):
	return -2.0 * ((eps + (math.pow(eps, 3.0) / 3.0)) + (math.pow(eps, 5.0) / 5.0))

Julia

function code(eps)
	return Float64(-2.0 * Float64(Float64(eps + Float64((eps ^ 3.0) / 3.0)) + Float64((eps ^ 5.0) / 5.0)))
end

MATLAB

function tmp = code(eps)
	tmp = -2.0 * ((eps + ((eps ^ 3.0) / 3.0)) + ((eps ^ 5.0) / 5.0));
end

Wolfram

code[eps_] := N[(-2.0 * N[(N[(eps + N[(N[Power[eps, 3.0], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 5.0], $MachinePrecision] / 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

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