sintan (problem 3.4.5)

Percentage Accurate: 1.6% → 99.9%

Time: 19.6s

Alternatives: 5

Speedup: 218.0×

Specification

\[-0.4 \leq \varepsilon \land \varepsilon \leq 0.4\]

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 1.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))

C

double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

Java

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

Python

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

Julia

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

MATLAB

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

Wolfram

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00024107142857142857, -0.009642857142857142\right), 0.225\right), -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma
  (* eps eps)
  (fma
   eps
   (* eps (fma eps (* eps 0.00024107142857142857) -0.009642857142857142))
   0.225)
  -0.5))

C

double code(double eps) {
	return fma((eps * eps), fma(eps, (eps * fma(eps, (eps * 0.00024107142857142857), -0.009642857142857142)), 0.225), -0.5);
}

Julia

function code(eps)
	return fma(Float64(eps * eps), fma(eps, Float64(eps * fma(eps, Float64(eps * 0.00024107142857142857), -0.009642857142857142)), 0.225), -0.5)
end

Wolfram

code[eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(eps * N[(eps * 0.00024107142857142857), $MachinePrecision] + -0.009642857142857142), $MachinePrecision]), $MachinePrecision] + 0.225), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.7%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + {\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right) + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)\right)} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right), \frac{9}{40}\right)}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}\right)}, \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\frac{27}{112000} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)\right)}, \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\color{blue}{\left(\frac{27}{112000} \cdot \varepsilon\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(\frac{27}{112000} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\frac{27}{2800}\right)\right)\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{27}{112000} \cdot \varepsilon\right) + \color{blue}{\frac{-27}{2800}}\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{27}{112000} \cdot \varepsilon, \frac{-27}{2800}\right)}, \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{27}{112000}}, \frac{-27}{2800}\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{27}{112000}}, \frac{-27}{2800}\right), \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   18. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00024107142857142857, -0.009642857142857142\right), 0.225\right), \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00024107142857142857, -0.009642857142857142\right), 0.225\right), -0.5\right)} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.009642857142857142\right)\right), \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.225, -0.5\right)\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma
  (* eps (* eps (* eps -0.009642857142857142)))
  eps
  (fma eps (* eps 0.225) -0.5)))

C

double code(double eps) {
	return fma((eps * (eps * (eps * -0.009642857142857142))), eps, fma(eps, (eps * 0.225), -0.5));
}

Julia

function code(eps)
	return fma(Float64(eps * Float64(eps * Float64(eps * -0.009642857142857142))), eps, fma(eps, Float64(eps * 0.225), -0.5))
end

Wolfram

code[eps_] := N[(N[(eps * N[(eps * N[(eps * -0.009642857142857142), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(eps * N[(eps * 0.225), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.7%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{-27}{2800} \cdot {\varepsilon}^{2} + \frac{9}{40}}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{-27}{2800}} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-27}{2800} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-27}{2800}\right)} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-27}{2800}, \frac{9}{40}\right)}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{-27}{2800}}, \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   11. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.009642857142857142, 0.225\right), \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.009642857142857142, 0.225\right), -0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot -0.009642857142857142\right)\right) \cdot \varepsilon, \color{blue}{\varepsilon}, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.225, -0.5\right)\right) \]

8. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.009642857142857142\right)\right), \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.225, -0.5\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.009642857142857142, 0.225\right), -0.5\right) \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (fma (* eps eps) (fma eps (* eps -0.009642857142857142) 0.225) -0.5))

C

double code(double eps) {
	return fma((eps * eps), fma(eps, (eps * -0.009642857142857142), 0.225), -0.5);
}

Julia

function code(eps)
	return fma(Float64(eps * eps), fma(eps, Float64(eps * -0.009642857142857142), 0.225), -0.5)
end

Wolfram

code[eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * -0.009642857142857142), $MachinePrecision] + 0.225), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.7%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{9}{40} + \frac{-27}{2800} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{-27}{2800} \cdot {\varepsilon}^{2} + \frac{9}{40}}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{-27}{2800}} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-27}{2800} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-27}{2800}\right)} + \frac{9}{40}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-27}{2800}, \frac{9}{40}\right)}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{-27}{2800}}, \frac{9}{40}\right), \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   11. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.009642857142857142, 0.225\right), \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.009642857142857142, 0.225\right), -0.5\right)} \]
6. Add Preprocessing

Alternative 4: 99.6% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.225, \varepsilon \cdot \varepsilon, -0.5\right) \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (fma 0.225 (* eps eps) -0.5))

C

double code(double eps) {
	return fma(0.225, (eps * eps), -0.5);
}

Julia

function code(eps)
	return fma(0.225, Float64(eps * eps), -0.5)
end

Wolfram

code[eps_] := N[(0.225 * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]

Derivation

1. Initial program 1.7%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{9}{40}, {\varepsilon}^{2}, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{9}{40}, \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{9}{40}, \color{blue}{\varepsilon \cdot \varepsilon}, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. metadata-eval 99.8

      \[\leadsto \mathsf{fma}\left(0.225, \varepsilon \cdot \varepsilon, \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, \varepsilon \cdot \varepsilon, -0.5\right)} \]
6. Add Preprocessing

Alternative 5: 99.1% accurate, 218.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 -0.5)

C

double code(double eps) {
	return -0.5;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = -0.5d0
end function

Java

public static double code(double eps) {
	return -0.5;
}

Python

def code(eps):
	return -0.5

Julia

function code(eps)
	return -0.5
end

MATLAB

function tmp = code(eps)
	tmp = -0.5;
end

Wolfram

code[eps_] := -0.5

Derivation

1. Initial program 1.7%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{-1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{-0.5} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \left(\left(-0.5 + \frac{9 \cdot \left(\varepsilon \cdot \varepsilon\right)}{40}\right) + \frac{-27 \cdot t\_0}{2800}\right) + \frac{27 \cdot \left(\left(t\_0 \cdot \varepsilon\right) \cdot \varepsilon\right)}{112000} \end{array} \end{array} \]

FPCore

(FPCore (eps)
 :precision binary64
 (let* ((t_0 (* (* (* eps eps) eps) eps)))
   (+
    (+ (+ -0.5 (/ (* 9.0 (* eps eps)) 40.0)) (/ (* -27.0 t_0) 2800.0))
    (/ (* 27.0 (* (* t_0 eps) eps)) 112000.0))))

C

double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = ((eps * eps) * eps) * eps
    code = (((-0.5d0) + ((9.0d0 * (eps * eps)) / 40.0d0)) + (((-27.0d0) * t_0) / 2800.0d0)) + ((27.0d0 * ((t_0 * eps) * eps)) / 112000.0d0)
end function

Java

public static double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

Python

def code(eps):
	t_0 = ((eps * eps) * eps) * eps
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0)

Julia

function code(eps)
	t_0 = Float64(Float64(Float64(eps * eps) * eps) * eps)
	return Float64(Float64(Float64(-0.5 + Float64(Float64(9.0 * Float64(eps * eps)) / 40.0)) + Float64(Float64(-27.0 * t_0) / 2800.0)) + Float64(Float64(27.0 * Float64(Float64(t_0 * eps) * eps)) / 112000.0))
end

MATLAB

function tmp = code(eps)
	t_0 = ((eps * eps) * eps) * eps;
	tmp = ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
end

Wolfram

code[eps_] := Block[{t$95$0 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, N[(N[(N[(-0.5 + N[(N[(9.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / 40.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-27.0 * t$95$0), $MachinePrecision] / 2800.0), $MachinePrecision]), $MachinePrecision] + N[(N[(27.0 * N[(N[(t$95$0 * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] / 112000.0), $MachinePrecision]), $MachinePrecision]]

Developer Target 2: 99.6% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \left(0.225 \cdot \varepsilon\right) \cdot \varepsilon - 0.5 \end{array} \]

FPCore

(FPCore (eps) :precision binary64 (- (* (* 0.225 eps) eps) 0.5))

C

double code(double eps) {
	return ((0.225 * eps) * eps) - 0.5;
}

Fortran

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = ((0.225d0 * eps) * eps) - 0.5d0
end function

Java

public static double code(double eps) {
	return ((0.225 * eps) * eps) - 0.5;
}

Python

def code(eps):
	return ((0.225 * eps) * eps) - 0.5

Julia

function code(eps)
	return Float64(Float64(Float64(0.225 * eps) * eps) - 0.5)
end

MATLAB

function tmp = code(eps)
	tmp = ((0.225 * eps) * eps) - 0.5;
end

Wolfram

code[eps_] := N[(N[(N[(0.225 * eps), $MachinePrecision] * eps), $MachinePrecision] - 0.5), $MachinePrecision]

Reproduce

herbie shell --seed 2024232 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))

  :alt
  (! :herbie-platform default (+ -1/2 (/ (* 9 (* eps eps)) 40) (/ (* -27 (* eps eps eps eps)) 2800) (/ (* 27 (* eps eps eps eps eps eps)) 112000)))

  :alt
  (! :herbie-platform default (- (* 9/40 eps eps) 1/2))

  (/ (- eps (sin eps)) (- eps (tan eps))))