logq (problem 3.4.3)

Percentage Accurate: 8.5% → 100.0%

Time: 9.4s

Alternatives: 7

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: 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]

Derivation

1. Initial program 8.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sub-neg N/A

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

   5. remove-double-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. neg-sub0 N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\color{blue}{0 - \left(1 - \varepsilon\right)}}}\right) \]

   16. lift--.f64 N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{0 - \color{blue}{\left(1 - \varepsilon\right)}}}\right) \]

   17. associate--r- N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\color{blue}{\left(0 - 1\right) + \varepsilon}}}\right) \]

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 8.0

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

4. Applied rewrites 8.0%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. log-div N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lift-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-+.f64 N/A

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

   13. flip-- N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. log-div N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eps_] := N[(eps * N[(eps * N[(eps * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * -0.2857142857142857 + -0.4), $MachinePrecision]), $MachinePrecision] + -0.6666666666666666), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 3: 99.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sub-neg N/A

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

   5. remove-double-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. neg-sub0 N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\color{blue}{0 - \left(1 - \varepsilon\right)}}}\right) \]

   16. lift--.f64 N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{0 - \color{blue}{\left(1 - \varepsilon\right)}}}\right) \]

   17. associate--r- N/A

       \[\leadsto \log \left(\frac{1 - \varepsilon}{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\color{blue}{\left(0 - 1\right) + \varepsilon}}}\right) \]

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 8.0

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

4. Applied rewrites 8.0%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.6%

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

Alternative 4: 99.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eps_] := N[(eps * N[(eps * N[(eps * N[(eps * N[(eps * -0.4), $MachinePrecision] + -0.6666666666666666), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eps_] := N[(eps * N[(eps * N[(eps * -0.6666666666666666), $MachinePrecision] + -2.0), $MachinePrecision]), $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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 98.9% accurate, 19.7× 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 8.2%

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right) \]
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. Final simplification 99.1%

   \[\leadsto \varepsilon \cdot -2 \]
7. Add Preprocessing

Alternative 7: 5.3% accurate, 118.0× 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 8.2%

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

4. Applied rewrites 5.5%

   \[\leadsto \color{blue}{0} \]
5. 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 2024227 
(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))))