logq (problem 3.4.3)

Percentage Accurate: 8.0% → 99.8%

Time: 7.5s

Alternatives: 4

Speedup: 19.7×

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: 8.0% 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: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(-0.2857142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.4\right) \cdot \varepsilon\right) \cdot \varepsilon - 0.6666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 2\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (*
  (-
   (*
    (-
     (* (* (- (* -0.2857142857142857 (* eps eps)) 0.4) eps) eps)
     0.6666666666666666)
    (* eps eps))
   2.0)
  eps))

C

double code(double eps) {
	return (((((((-0.2857142857142857 * (eps * eps)) - 0.4) * eps) * eps) - 0.6666666666666666) * (eps * eps)) - 2.0) * eps;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = ((((((((-0.2857142857142857d0) * (eps * eps)) - 0.4d0) * eps) * eps) - 0.6666666666666666d0) * (eps * eps)) - 2.0d0) * eps
end function

Java

public static double code(double eps) {
	return (((((((-0.2857142857142857 * (eps * eps)) - 0.4) * eps) * eps) - 0.6666666666666666) * (eps * eps)) - 2.0) * eps;
}

Python

def code(eps):
	return (((((((-0.2857142857142857 * (eps * eps)) - 0.4) * eps) * eps) - 0.6666666666666666) * (eps * eps)) - 2.0) * eps

Julia

function code(eps)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.2857142857142857 * Float64(eps * eps)) - 0.4) * eps) * eps) - 0.6666666666666666) * Float64(eps * eps)) - 2.0) * eps)
end

MATLAB

function tmp = code(eps)
	tmp = (((((((-0.2857142857142857 * (eps * eps)) - 0.4) * eps) * eps) - 0.6666666666666666) * (eps * eps)) - 2.0) * eps;
end

Wolfram

code[eps_] := N[(N[(N[(N[(N[(N[(N[(N[(-0.2857142857142857 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.4), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{7} \cdot {\varepsilon}^{2} - \frac{2}{5}\right) - \frac{2}{3}\right) - 2\right) \cdot \varepsilon} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(\left(\left(-0.2857142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.4\right) \cdot \varepsilon\right) \cdot \varepsilon - 0.6666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 2\right) \cdot \varepsilon} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right) \cdot \varepsilon} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right)} \cdot \varepsilon \]

   4. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) - 2\right) \cdot \varepsilon \]

   5. associate-*l* N/A

      \[\leadsto \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right)} - 2\right) \cdot \varepsilon \]

   6. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right) \cdot \varepsilon} - 2\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)\right) \cdot \varepsilon} - 2\right) \cdot \varepsilon \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) \cdot \varepsilon\right)} \cdot \varepsilon - 2\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right) \cdot \varepsilon\right)} \cdot \varepsilon - 2\right) \cdot \varepsilon \]

   10. lower--.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\left(\frac{-2}{5} \cdot {\varepsilon}^{2} - \frac{2}{3}\right)} \cdot \varepsilon\right) \cdot \varepsilon - 2\right) \cdot \varepsilon \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(\color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{2}} - \frac{2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon - 2\right) \cdot \varepsilon \]

   12. unpow2 N/A

       \[\leadsto \left(\left(\left(\frac{-2}{5} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{2}{3}\right) \cdot \varepsilon\right) \cdot \varepsilon - 2\right) \cdot \varepsilon \]

   13. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right) \cdot \varepsilon} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-2}{3} \cdot {\varepsilon}^{2} - 2\right)} \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{-2}{3}} - 2\right) \cdot \varepsilon \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{-2}{3}} - 2\right) \cdot \varepsilon \]

   6. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-2}{3} - 2\right) \cdot \varepsilon \]

   7. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 4: 99.2% accurate, 19.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eps):
	return -2.0 * eps

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]
4. Step-by-step derivation

   1. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{-2 \cdot \varepsilon} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× 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 2024313 
(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))))