sintan (problem 3.4.5)

Percentage Accurate: 1.5% → 99.9%

Time: 14.8s

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: 1.5% 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, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 1.3%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate--l+ N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. pow-sqr N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{4}}, \frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2}} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{9}{40} - \frac{1}{2}\right) \]

   18. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 0.225 - 0.5\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.225 - 0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 1.3%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate--l+ N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. pow-sqr N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{4}}, \frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2}} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{9}{40} - \frac{1}{2}\right) \]

   18. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 0.225 - 0.5\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.225 - 0.5\right)} \]

6. Taylor expanded in eps around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 1.3%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate--l+ N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. pow-sqr N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\varepsilon}^{\color{blue}{4}}, \frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2} - \frac{27}{2800}}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \color{blue}{\frac{27}{112000} \cdot {\varepsilon}^{2}} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{27}{2800}, \frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \]

   14. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2}\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, \frac{27}{112000} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{27}{2800}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{9}{40} - \frac{1}{2}\right) \]

   18. lower-*.f64 99.9

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 0.225 - 0.5\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{4}, 0.00024107142857142857 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.009642857142857142, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.225 - 0.5\right)} \]

6. Taylor expanded in eps around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 15.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 1.3%

   \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
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. lower--.f64 N/A

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {\varepsilon}^{2} - \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \frac{9}{40}} - \frac{1}{2} \]

   4. unpow2 N/A

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{9}{40} - \frac{1}{2} \]

   5. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 5: 99.2% 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 1.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.1%

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

Developer Target 1: 99.7% 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 2024339 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))

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

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