logq (problem 3.4.3)

Percentage Accurate: 8.5% → 99.7%

Time: 7.3s

Alternatives: 6

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: 8.5% 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: 99.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-2}{5} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-2}{5} \cdot \varepsilon\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \]

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

7. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.5%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. sub-neg N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-2}{5} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-2}{5} \cdot \varepsilon\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon \]

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.4 \cdot \varepsilon, \varepsilon, -0.6666666666666666\right) \cdot \varepsilon, \varepsilon, -2\right) \cdot \varepsilon} \]
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 2024234 
(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))))