logq (problem 3.4.3)

Percentage Accurate: 8.5% → 100.0%

Time: 5.5s

Alternatives: 4

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: 8.5% 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 9.7%

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

   1. log-div 9.6%

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

   2. sub-neg 9.6%

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

   3. log1p-def 21.9%

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

   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, 9.7× speedup

Math

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

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(eps * Float64(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[(eps * N[(eps * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.7%

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

2. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{-2 \cdot \varepsilon + -0.6666666666666666 \cdot {\varepsilon}^{3}} \]
3. Step-by-step derivation

   1. +-commutative 99.4%

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

   2. unpow3 99.4%

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

   3. unpow2 99.4%

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

   4. associate-*r* 99.4%

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

   5. distribute-rgt-out 99.4%

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

   6. *-commutative 99.4%

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

   7. unpow2 99.4%

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

   8. associate-*l* 99.4%

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

   9. fma-def 99.4%

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

4. Simplified 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.6666666666666666, -2\right)} \]
5. Step-by-step derivation

   1. fma-udef 99.4%

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

   2. distribute-rgt-in 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.6666666666666666\right)\right) + \varepsilon \cdot -2 \]

Alternative 3: 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 9.7%

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

2. Taylor expanded in eps around 0 98.9%

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

3. Final simplification 98.9%

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

Alternative 4: 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 9.7%

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

   1. div-inv 9.7%

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

   2. sub-neg 9.7%

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

   3. add-sqr-sqrt 5.1%

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

   4. sqrt-unprod 7.9%

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

   5. sqr-neg 7.9%

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

   6. sqrt-unprod 2.8%

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

   7. add-sqr-sqrt 5.4%

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

   8. pow1 5.4%

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

   9. inv-pow 5.4%

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

   10. pow-prod-up 5.4%

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

   11. metadata-eval 5.4%

       \[\leadsto \log \left({\left(1 + \varepsilon\right)}^{\color{blue}{0}}\right) \]

   12. metadata-eval 5.4%

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

   13. metadata-eval 5.4%

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

3. Applied egg-rr 5.4%

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

4. Final simplification 5.4%

   \[\leadsto 0 \]

Developer target: 99.7% 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 2023187 
(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))))