2sin (example 3.3)

Percentage Accurate: 62.6% → 99.8%

Time: 14.3s

Alternatives: 7

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.00026041666666666666\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + 0.5 \cdot \varepsilon\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (+
    0.5
    (*
     (* eps eps)
     (+ -0.020833333333333332 (* (* eps eps) 0.00026041666666666666))))
   (* eps (cos (+ x (* 0.5 eps)))))
  2.0))

C

double code(double x, double eps) {
	return ((0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))) * (eps * cos((x + (0.5 * eps))))) * 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((0.5d0 + ((eps * eps) * ((-0.020833333333333332d0) + ((eps * eps) * 0.00026041666666666666d0)))) * (eps * cos((x + (0.5d0 * eps))))) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	return ((0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))) * (eps * Math.cos((x + (0.5 * eps))))) * 2.0;
}

Python

def code(x, eps):
	return ((0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))) * (eps * math.cos((x + (0.5 * eps))))) * 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(0.5 + Float64(Float64(eps * eps) * Float64(-0.020833333333333332 + Float64(Float64(eps * eps) * 0.00026041666666666666)))) * Float64(eps * cos(Float64(x + Float64(0.5 * eps))))) * 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))) * (eps * cos((x + (0.5 * eps))))) * 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.5 + N[(N[(eps * eps), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(eps * eps), $MachinePrecision] * 0.00026041666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[Cos[N[(x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   3. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   4. associate-*l* N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)}\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)}\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)}\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   7. sub-neg N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   9. +-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   11. *-commutative N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{48} + \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}}\right)\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{48} + \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}}\right)\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   13. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{48} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{3840}\right)\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   14. *-lowering-*.f64 100.0

       \[\leadsto \left(\left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.020833333333333332 + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot 0.00026041666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

7. Simplified 100.0%

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.00026041666666666666\right)\right)\right)\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right)} \cdot 2 \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right) \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)} \cdot 2 \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right)} \cdot 2 \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right)} \cdot 2 \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)} \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{2} + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)}\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   6. unpow2 N/A

      \[\leadsto \left(\left(\frac{1}{2} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{2} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   8. sub-neg N/A

      \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)}\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-1}{48}}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)}\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)}\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   14. unpow2 N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{3840}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{3840}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right) \cdot 2 \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3840}\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)}\right) \cdot 2 \]

   17. +-commutative N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3840}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right)\right)\right) \cdot 2 \]

   18. distribute-rgt-in N/A

       \[\leadsto \left(\left(\frac{1}{2} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{-1}{48} + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3840}\right)\right) \cdot \left(\varepsilon \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot 2 \]

10. Simplified 100.0%

    \[\leadsto \color{blue}{\left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.00026041666666666666\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)\right)} \cdot 2 \]

11. Final simplification 100.0%

    \[\leadsto \left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.00026041666666666666\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + 0.5 \cdot \varepsilon\right)\right)\right) \cdot 2 \]
12. Add Preprocessing

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\cos \left(x + 0.5 \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (cos (+ x (* 0.5 eps)))
   (* eps (+ 0.5 (* eps (* eps -0.020833333333333332)))))))

C

double code(double x, double eps) {
	return 2.0 * (cos((x + (0.5 * eps))) * (eps * (0.5 + (eps * (eps * -0.020833333333333332)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * (cos((x + (0.5d0 * eps))) * (eps * (0.5d0 + (eps * (eps * (-0.020833333333333332d0))))))
end function

Java

public static double code(double x, double eps) {
	return 2.0 * (Math.cos((x + (0.5 * eps))) * (eps * (0.5 + (eps * (eps * -0.020833333333333332)))));
}

Python

def code(x, eps):
	return 2.0 * (math.cos((x + (0.5 * eps))) * (eps * (0.5 + (eps * (eps * -0.020833333333333332)))))

Julia

function code(x, eps)
	return Float64(2.0 * Float64(cos(Float64(x + Float64(0.5 * eps))) * Float64(eps * Float64(0.5 + Float64(eps * Float64(eps * -0.020833333333333332))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * (cos((x + (0.5 * eps))) * (eps * (0.5 + (eps * (eps * -0.020833333333333332)))));
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(0.5 + N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2} \]

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)} \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x\right)\right)\right) \cdot 2 \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon - -2 \cdot x\right)}\right)\right) \cdot 2 \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)} \cdot 2 \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right) \cdot 2 \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right) \cdot 2 \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)}\right) \cdot 2 \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}\right)\right) \cdot 2 \]

   10. metadata-eval N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{2} \cdot x\right)\right)\right) \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right)\right) \cdot 2 \]

   12. distribute-rgt-in N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)}\right) \cdot 2 \]

   13. *-commutative N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\color{blue}{\left(x \cdot 2\right)} \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   14. associate-*l* N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\color{blue}{x \cdot \left(2 \cdot \frac{1}{2}\right)} + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   15. metadata-eval N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(x \cdot \color{blue}{1} + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   16. *-rgt-identity N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(\color{blue}{x} + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   17. *-commutative N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(x + \color{blue}{\frac{1}{2} \cdot \varepsilon}\right)\right) \cdot 2 \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   19. *-commutative N/A

       \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(x + \color{blue}{\varepsilon \cdot \frac{1}{2}}\right)\right) \cdot 2 \]

   20. *-lowering-*.f64 100.0

       \[\leadsto \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos \left(x + \color{blue}{\varepsilon \cdot 0.5}\right)\right) \cdot 2 \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)} \cdot 2 \]

8. Taylor expanded in eps around 0

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)} \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)} \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)}\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{48}}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   4. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-1}{48}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   5. associate-*l* N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{48}\right)}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \color{blue}{\left(\frac{-1}{48} \cdot \varepsilon\right)}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \color{blue}{\varepsilon \cdot \left(\frac{-1}{48} \cdot \varepsilon\right)}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   8. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{-1}{48}\right)}\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2 \]

   9. *-lowering-*.f64 99.9

      \[\leadsto \left(\left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.020833333333333332\right)}\right)\right) \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right) \cdot 2 \]

10. Simplified 99.9%

    \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right) \cdot 2 \]

11. Final simplification 99.9%

    \[\leadsto 2 \cdot \left(\cos \left(x + 0.5 \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \left(0.5 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos \left(\frac{1}{\frac{1}{x + 0.5 \cdot \varepsilon}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (cos (/ 1.0 (/ 1.0 (+ x (* 0.5 eps)))))))

C

double code(double x, double eps) {
	return eps * cos((1.0 / (1.0 / (x + (0.5 * eps)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * cos((1.0d0 / (1.0d0 / (x + (0.5d0 * eps)))))
end function

Java

public static double code(double x, double eps) {
	return eps * Math.cos((1.0 / (1.0 / (x + (0.5 * eps)))));
}

Python

def code(x, eps):
	return eps * math.cos((1.0 / (1.0 / (x + (0.5 * eps)))))

Julia

function code(x, eps)
	return Float64(eps * cos(Float64(1.0 / Float64(1.0 / Float64(x + Float64(0.5 * eps))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * cos((1.0 / (1.0 / (x + (0.5 * eps)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[Cos[N[(1.0 / N[(1.0 / N[(x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2} \]

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   2. *-lowering-*.f64 99.7

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

7. Simplified 99.7%

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

8. Taylor expanded in eps around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{\left(x \cdot 2\right)} \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right) \]

   5. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{x \cdot \left(2 \cdot \frac{1}{2}\right)} + \varepsilon \cdot \frac{1}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \cos \left(x \cdot \color{blue}{1} + \varepsilon \cdot \frac{1}{2}\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{x} + \varepsilon \cdot \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\frac{1}{2} \cdot \varepsilon}\right) \]

   9. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \varepsilon\right) \]

   10. cancel-sign-sub-inv N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cancel-sign-sub-inv N/A

       \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon\right)} \]

   13. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\frac{1}{2}} \cdot \varepsilon\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \]

   15. *-commutative N/A

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\varepsilon \cdot \frac{1}{2}}\right) \]

   16. *-lowering-*.f64 99.7

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\varepsilon \cdot 0.5}\right) \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)} \]
11. Step-by-step derivation

    1. flip-+ N/A

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

    2. clear-num N/A

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

    3. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \cos \left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{x - \varepsilon \cdot \frac{1}{2}}{x \cdot x - \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)}}\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(\frac{\mathsf{neg}\left(-1\right)}{\frac{x - \varepsilon \cdot \frac{1}{2}}{x \cdot x - \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)}}\right)} \]

    5. metadata-eval N/A

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

    6. clear-num N/A

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

    7. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{x \cdot x - \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \frac{1}{2}\right)}{x - \varepsilon \cdot \frac{1}{2}}}}\right) \]

    8. flip-+ N/A

       \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{x + \varepsilon \cdot \frac{1}{2}}}}\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{x + \varepsilon \cdot \frac{1}{2}}}}\right) \]

    10. metadata-eval N/A

        \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{\color{blue}{1}}{x + \varepsilon \cdot \frac{1}{2}}}\right) \]

    11. +-lowering-+.f64 N/A

        \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{1}{\color{blue}{x + \varepsilon \cdot \frac{1}{2}}}}\right) \]

    12. *-commutative N/A

        \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{1}{x + \color{blue}{\frac{1}{2} \cdot \varepsilon}}}\right) \]

    13. *-lowering-*.f64 99.7

        \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{\frac{1}{x + \color{blue}{0.5 \cdot \varepsilon}}}\right) \]

12. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(\frac{1}{\frac{1}{x + 0.5 \cdot \varepsilon}}\right)} \]
13. Add Preprocessing

Alternative 4: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos \left(x + 0.5 \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (cos (+ x (* 0.5 eps)))))

C

double code(double x, double eps) {
	return eps * cos((x + (0.5 * eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * cos((x + (0.5d0 * eps)))
end function

Java

public static double code(double x, double eps) {
	return eps * Math.cos((x + (0.5 * eps)));
}

Python

def code(x, eps):
	return eps * math.cos((x + (0.5 * eps)))

Julia

function code(x, eps)
	return Float64(eps * cos(Float64(x + Float64(0.5 * eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * cos((x + (0.5 * eps)));
end

Wolfram

code[x_, eps_] := N[(eps * N[Cos[N[(x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2} \]

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

   2. *-lowering-*.f64 99.7

      \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

7. Simplified 99.7%

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

8. Taylor expanded in eps around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{\left(x \cdot 2\right)} \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right) \]

   5. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{x \cdot \left(2 \cdot \frac{1}{2}\right)} + \varepsilon \cdot \frac{1}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \cos \left(x \cdot \color{blue}{1} + \varepsilon \cdot \frac{1}{2}\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \varepsilon \cdot \cos \left(\color{blue}{x} + \varepsilon \cdot \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\frac{1}{2} \cdot \varepsilon}\right) \]

   9. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \varepsilon\right) \]

   10. cancel-sign-sub-inv N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cancel-sign-sub-inv N/A

       \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon\right)} \]

   13. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\frac{1}{2}} \cdot \varepsilon\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \varepsilon \cdot \cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \]

   15. *-commutative N/A

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\varepsilon \cdot \frac{1}{2}}\right) \]

   16. *-lowering-*.f64 99.7

       \[\leadsto \varepsilon \cdot \cos \left(x + \color{blue}{\varepsilon \cdot 0.5}\right) \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)} \]

11. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \cos \left(x + 0.5 \cdot \varepsilon\right) \]
12. Add Preprocessing

Alternative 5: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (cos x)))

C

double code(double x, double eps) {
	return eps * cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * cos(x)
end function

Java

public static double code(double x, double eps) {
	return eps * Math.cos(x);
}

Python

def code(x, eps):
	return eps * math.cos(x)

Julia

function code(x, eps)
	return Float64(eps * cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * cos(x);
end

Wolfram

code[x_, eps_] := N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]

   2. cos-lowering-cos.f64 99.3

      \[\leadsto \varepsilon \cdot \color{blue}{\cos x} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
6. Add Preprocessing

Alternative 6: 97.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* (* eps eps) -0.16666666666666666))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((eps * eps) * (-0.16666666666666666d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666));
}

Python

def code(x, eps):
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * -0.16666666666666666)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((eps * eps) * -0.16666666666666666));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. sin-lowering-sin.f64 98.1

      \[\leadsto \color{blue}{\sin \varepsilon} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

6. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {\varepsilon}^{2}}\right) \]

   4. unpow2 N/A

      \[\leadsto \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   5. *-lowering-*.f64 98.1

      \[\leadsto \varepsilon \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

8. Simplified 98.1%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

9. Final simplification 98.1%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 7: 97.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. sin-lowering-sin.f64 98.1

      \[\leadsto \color{blue}{\sin \varepsilon} \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon} \]
7. Step-by-step derivation

8. Simplified 98.1%

   \[\leadsto \color{blue}{\varepsilon} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \cos \left(x + \frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* 2.0 (cos (+ x (/ eps 2.0)))) (sin (/ eps 2.0))))

C

double code(double x, double eps) {
	return (2.0 * cos((x + (eps / 2.0)))) * sin((eps / 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 * cos((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return (2.0 * Math.cos((x + (eps / 2.0)))) * Math.sin((eps / 2.0));
}

Python

def code(x, eps):
	return (2.0 * math.cos((x + (eps / 2.0)))) * math.sin((eps / 2.0))

Julia

function code(x, eps)
	return Float64(Float64(2.0 * cos(Float64(x + Float64(eps / 2.0)))) * sin(Float64(eps / 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 * cos((x + (eps / 2.0)))) * sin((eps / 2.0));
end

Wolfram

code[x_, eps_] := N[(N[(2.0 * N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (sin x) (- (cos eps) 1.0)) (* (cos x) (sin eps))))

C

double code(double x, double eps) {
	return (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(x) * (cos(eps) - 1.0d0)) + (cos(x) * sin(eps))
end function

Java

public static double code(double x, double eps) {
	return (Math.sin(x) * (Math.cos(eps) - 1.0)) + (Math.cos(x) * Math.sin(eps));
}

Python

def code(x, eps):
	return (math.sin(x) * (math.cos(eps) - 1.0)) + (math.cos(x) * math.sin(eps))

Julia

function code(x, eps)
	return Float64(Float64(sin(x) * Float64(cos(eps) - 1.0)) + Float64(cos(x) * sin(eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(eps));
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Developer Target 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (cos (* 0.5 (- eps (* -2.0 x)))) (sin (* 0.5 eps))) 2.0))

C

double code(double x, double eps) {
	return (cos((0.5 * (eps - (-2.0 * x)))) * sin((0.5 * eps))) * 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos((0.5d0 * (eps - ((-2.0d0) * x)))) * sin((0.5d0 * eps))) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (Math.cos((0.5 * (eps - (-2.0 * x)))) * Math.sin((0.5 * eps))) * 2.0;
}

Python

def code(x, eps):
	return (math.cos((0.5 * (eps - (-2.0 * x)))) * math.sin((0.5 * eps))) * 2.0

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(0.5 * Float64(eps - Float64(-2.0 * x)))) * sin(Float64(0.5 * eps))) * 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos((0.5 * (eps - (-2.0 * x)))) * sin((0.5 * eps))) * 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(0.5 * N[(eps - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Reproduce

herbie shell --seed 2024191 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

  (- (sin (+ x eps)) (sin x)))