logq (problem 3.4.3)

Percentage Accurate: 8.5% → 100.0%

Time: 9.2s

Alternatives: 11

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1, \mathsf{log1p}\left(\varepsilon\right), \mathsf{log1p}\left(\left(-\varepsilon\right) \cdot \varepsilon\right)\right) - \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (- (fma -1.0 (log1p eps) (log1p (* (- eps) eps))) (log1p eps)))

C

double code(double eps) {
	return fma(-1.0, log1p(eps), log1p((-eps * eps))) - log1p(eps);
}

Julia

function code(eps)
	return Float64(fma(-1.0, log1p(eps), log1p(Float64(Float64(-eps) * eps))) - log1p(eps))
end

Wolfram

code[eps_] := N[(N[(-1.0 * N[Log[1 + eps], $MachinePrecision] + N[Log[1 + N[((-eps) * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[1 + eps], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-+.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. flip-- N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 9.9%

   \[\leadsto \log \left(\frac{\color{blue}{\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}}}{1 + \varepsilon}\right) \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \color{blue}{\log \left(\frac{\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}}{1 + \varepsilon}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \log \color{blue}{\left(\frac{\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}}{1 + \varepsilon}\right)} \]

   3. log-div N/A

      \[\leadsto \color{blue}{\log \left(\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}\right) - \log \left(1 + \varepsilon\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \log \left(\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}\right) - \log \color{blue}{\left(1 + \varepsilon\right)} \]

   5. lift-log1p.f64 N/A

      \[\leadsto \log \left(\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}\right) - \color{blue}{\mathsf{log1p}\left(\varepsilon\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\log \left(\frac{\left(1 - {\varepsilon}^{4}\right) \cdot e^{-\mathsf{log1p}\left(\varepsilon\right)}}{\mathsf{fma}\left(\varepsilon, \varepsilon, 1\right)}\right) - \mathsf{log1p}\left(\varepsilon\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \mathsf{log1p}\left(\varepsilon\right), \mathsf{log1p}\left(\left(-\varepsilon\right) \cdot \varepsilon\right)\right) - \mathsf{log1p}\left(\varepsilon\right)} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.02328042328042328, \varepsilon \cdot \varepsilon, 0.044444444444444446\right), \varepsilon \cdot \varepsilon, 0.16666666666666666\right), \varepsilon \cdot \varepsilon, -0.5\right)} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/
  eps
  (fma
   (fma
    (fma 0.02328042328042328 (* eps eps) 0.044444444444444446)
    (* eps eps)
    0.16666666666666666)
   (* eps eps)
   -0.5)))

C

double code(double eps) {
	return eps / fma(fma(fma(0.02328042328042328, (eps * eps), 0.044444444444444446), (eps * eps), 0.16666666666666666), (eps * eps), -0.5);
}

Julia

function code(eps)
	return Float64(eps / fma(fma(fma(0.02328042328042328, Float64(eps * eps), 0.044444444444444446), Float64(eps * eps), 0.16666666666666666), Float64(eps * eps), -0.5))
end

Wolfram

code[eps_] := N[(eps / N[(N[(N[(0.02328042328042328 * N[(eps * eps), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.9%

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

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right), \varepsilon \cdot \varepsilon, -2\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon}{{\varepsilon}^{2} \cdot \left(\frac{1}{6} + {\varepsilon}^{2} \cdot \left(\frac{2}{45} + \frac{22}{945} \cdot {\varepsilon}^{2}\right)\right) - \color{blue}{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.02328042328042328, \varepsilon \cdot \varepsilon, 0.044444444444444446\right), \varepsilon \cdot \varepsilon, 0.16666666666666666\right), \color{blue}{\varepsilon \cdot \varepsilon}, -0.5\right)} \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

Alternative 4: 99.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, \varepsilon \cdot \varepsilon, 0.16666666666666666\right), \varepsilon \cdot \varepsilon, -0.5\right)} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (/
  eps
  (fma
   (fma 0.044444444444444446 (* eps eps) 0.16666666666666666)
   (* eps eps)
   -0.5)))

C

double code(double eps) {
	return eps / fma(fma(0.044444444444444446, (eps * eps), 0.16666666666666666), (eps * eps), -0.5);
}

Julia

function code(eps)
	return Float64(eps / fma(fma(0.044444444444444446, Float64(eps * eps), 0.16666666666666666), Float64(eps * eps), -0.5))
end

Wolfram

code[eps_] := N[(eps / N[(N[(0.044444444444444446 * N[(eps * eps), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.9%

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

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right), \varepsilon \cdot \varepsilon, -2\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon}{{\varepsilon}^{2} \cdot \left(\frac{1}{6} + \frac{2}{45} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, \varepsilon \cdot \varepsilon, 0.16666666666666666\right), \color{blue}{\varepsilon \cdot \varepsilon}, -0.5\right)} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

Alternative 6: 99.6% accurate, 4.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 7: 99.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(0.16666666666666666, \varepsilon \cdot \varepsilon, -0.5\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.2857142857142857, -0.4\right), \varepsilon \cdot \varepsilon, -0.6666666666666666\right), \varepsilon \cdot \varepsilon, -2\right)}}} \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon}{\frac{1}{6} \cdot {\varepsilon}^{2} - \color{blue}{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{\varepsilon \cdot \varepsilon}, -0.5\right)} \]
11. Add Preprocessing

Alternative 8: 99.4% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 9: 99.4% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.9%

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

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 10: 98.9% 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 9.9%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 98.0

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

5. Applied rewrites 98.0%

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

Alternative 11: 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 9.9%

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

4. Applied rewrites 5.0%

   \[\leadsto \color{blue}{0} \]
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 2024296 
(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))))