2cos (problem 3.3.5)

Percentage Accurate: 52.3% → 99.6%

Time: 25.2s

Alternatives: 13

Speedup: 51.3×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot 0.041666666666666664 + \varepsilon \cdot \left(-0.001388888888888889 \cdot \left(\varepsilon \cdot \cos x\right) - 0.008333333333333333 \cdot \sin x\right)\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (*
    eps
    (+
     (* -0.5 (cos x))
     (*
      eps
      (-
       (*
        eps
        (+
         (* (cos x) 0.041666666666666664)
         (*
          eps
          (-
           (* -0.001388888888888889 (* eps (cos x)))
           (* 0.008333333333333333 (sin x))))))
       (* (sin x) -0.16666666666666666)))))
   (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664) + (eps * ((-0.001388888888888889 * (eps * cos(x))) - (0.008333333333333333 * sin(x)))))) - (sin(x) * -0.16666666666666666))))) - sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (((-0.5d0) * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664d0) + (eps * (((-0.001388888888888889d0) * (eps * cos(x))) - (0.008333333333333333d0 * sin(x)))))) - (sin(x) * (-0.16666666666666666d0)))))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * Math.cos(x)) + (eps * ((eps * ((Math.cos(x) * 0.041666666666666664) + (eps * ((-0.001388888888888889 * (eps * Math.cos(x))) - (0.008333333333333333 * Math.sin(x)))))) - (Math.sin(x) * -0.16666666666666666))))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * ((-0.5 * math.cos(x)) + (eps * ((eps * ((math.cos(x) * 0.041666666666666664) + (eps * ((-0.001388888888888889 * (eps * math.cos(x))) - (0.008333333333333333 * math.sin(x)))))) - (math.sin(x) * -0.16666666666666666))))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(Float64(-0.5 * cos(x)) + Float64(eps * Float64(Float64(eps * Float64(Float64(cos(x) * 0.041666666666666664) + Float64(eps * Float64(Float64(-0.001388888888888889 * Float64(eps * cos(x))) - Float64(0.008333333333333333 * sin(x)))))) - Float64(sin(x) * -0.16666666666666666))))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * ((-0.5 * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664) + (eps * ((-0.001388888888888889 * (eps * cos(x))) - (0.008333333333333333 * sin(x)))))) - (sin(x) * -0.16666666666666666))))) - sin(x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + N[(eps * N[(N[(-0.001388888888888889 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.008333333333333333 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(0.041666666666666664 \cdot \cos x + \varepsilon \cdot \left(-0.001388888888888889 \cdot \left(\varepsilon \cdot \cos x\right) - 0.008333333333333333 \cdot \sin x\right)\right) - -0.16666666666666666 \cdot \sin x\right)\right) - \sin x\right)} \]

4. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot 0.041666666666666664 + \varepsilon \cdot \left(-0.001388888888888889 \cdot \left(\varepsilon \cdot \cos x\right) - 0.008333333333333333 \cdot \sin x\right)\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x + \varepsilon \cdot \left(\cos x \cdot 0.5 + \varepsilon \cdot \left(\sin x \cdot -0.125 + \varepsilon \cdot \left(\cos x \cdot -0.020833333333333332 + 0.0026041666666666665 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  -2.0
  (*
   (sin (* eps 0.5))
   (+
    (sin x)
    (*
     eps
     (+
      (* (cos x) 0.5)
      (*
       eps
       (+
        (* (sin x) -0.125)
        (*
         eps
         (+
          (* (cos x) -0.020833333333333332)
          (* 0.0026041666666666665 (* eps (sin x)))))))))))))

C

double code(double x, double eps) {
	return -2.0 * (sin((eps * 0.5)) * (sin(x) + (eps * ((cos(x) * 0.5) + (eps * ((sin(x) * -0.125) + (eps * ((cos(x) * -0.020833333333333332) + (0.0026041666666666665 * (eps * sin(x)))))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin((eps * 0.5d0)) * (sin(x) + (eps * ((cos(x) * 0.5d0) + (eps * ((sin(x) * (-0.125d0)) + (eps * ((cos(x) * (-0.020833333333333332d0)) + (0.0026041666666666665d0 * (eps * sin(x)))))))))))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin((eps * 0.5)) * (Math.sin(x) + (eps * ((Math.cos(x) * 0.5) + (eps * ((Math.sin(x) * -0.125) + (eps * ((Math.cos(x) * -0.020833333333333332) + (0.0026041666666666665 * (eps * Math.sin(x)))))))))));
}

Python

def code(x, eps):
	return -2.0 * (math.sin((eps * 0.5)) * (math.sin(x) + (eps * ((math.cos(x) * 0.5) + (eps * ((math.sin(x) * -0.125) + (eps * ((math.cos(x) * -0.020833333333333332) + (0.0026041666666666665 * (eps * math.sin(x)))))))))))

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps * 0.5)) * Float64(sin(x) + Float64(eps * Float64(Float64(cos(x) * 0.5) + Float64(eps * Float64(Float64(sin(x) * -0.125) + Float64(eps * Float64(Float64(cos(x) * -0.020833333333333332) + Float64(0.0026041666666666665 * Float64(eps * sin(x))))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((eps * 0.5)) * (sin(x) + (eps * ((cos(x) * 0.5) + (eps * ((sin(x) * -0.125) + (eps * ((cos(x) * -0.020833333333333332) + (0.0026041666666666665 * (eps * sin(x)))))))))));
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * 0.5), $MachinePrecision] + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * -0.125), $MachinePrecision] + N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * -0.020833333333333332), $MachinePrecision] + N[(0.0026041666666666665 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

   1. diff-cos 80.8%

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

   2. div-inv 80.8%

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

   3. associate--l+ 80.8%

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

   4. metadata-eval 80.8%

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

   5. div-inv 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   6. +-commutative 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(\color{blue}{\left(\varepsilon + x\right)} + x\right) \cdot \frac{1}{2}\right)\right) \]

   7. associate-+l+ 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(\varepsilon + \left(x + x\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   8. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r- 80.8%

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

   2. +-commutative 80.8%

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

   3. associate--l+ 99.5%

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

   4. +-inverses 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + \color{blue}{0}\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   5. +-commutative 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(0 + \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   6. *-lft-identity 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(0 + \color{blue}{1 \cdot \varepsilon}\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   7. metadata-eval 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(0 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   8. cancel-sign-sub-inv 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   9. neg-sub0 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(--1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   10. mul-1-neg 99.5%

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

   11. remove-double-neg 99.5%

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

   12. *-commutative 99.5%

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

   13. +-commutative 99.5%

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

   14. count-2 99.5%

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

   15. fma-define 99.5%

       \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)}\right)\right) \]

6. Simplified 99.5%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)} \]

7. Taylor expanded in eps around 0 99.6%

   \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\left(\sin x + \varepsilon \cdot \left(0.5 \cdot \cos x + \varepsilon \cdot \left(-0.125 \cdot \sin x + \varepsilon \cdot \left(-0.020833333333333332 \cdot \cos x + 0.0026041666666666665 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)\right)}\right) \]

8. Final simplification 99.6%

   \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x + \varepsilon \cdot \left(\cos x \cdot 0.5 + \varepsilon \cdot \left(\sin x \cdot -0.125 + \varepsilon \cdot \left(\cos x \cdot -0.020833333333333332 + 0.0026041666666666665 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot 0.041666666666666664 + \left(\varepsilon \cdot \sin x\right) \cdot -0.008333333333333333\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (*
    eps
    (+
     (* -0.5 (cos x))
     (*
      eps
      (-
       (*
        eps
        (+
         (* (cos x) 0.041666666666666664)
         (* (* eps (sin x)) -0.008333333333333333)))
       (* (sin x) -0.16666666666666666)))))
   (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664) + ((eps * sin(x)) * -0.008333333333333333))) - (sin(x) * -0.16666666666666666))))) - sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (((-0.5d0) * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664d0) + ((eps * sin(x)) * (-0.008333333333333333d0)))) - (sin(x) * (-0.16666666666666666d0)))))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * Math.cos(x)) + (eps * ((eps * ((Math.cos(x) * 0.041666666666666664) + ((eps * Math.sin(x)) * -0.008333333333333333))) - (Math.sin(x) * -0.16666666666666666))))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * ((-0.5 * math.cos(x)) + (eps * ((eps * ((math.cos(x) * 0.041666666666666664) + ((eps * math.sin(x)) * -0.008333333333333333))) - (math.sin(x) * -0.16666666666666666))))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(Float64(-0.5 * cos(x)) + Float64(eps * Float64(Float64(eps * Float64(Float64(cos(x) * 0.041666666666666664) + Float64(Float64(eps * sin(x)) * -0.008333333333333333))) - Float64(sin(x) * -0.16666666666666666))))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * ((-0.5 * cos(x)) + (eps * ((eps * ((cos(x) * 0.041666666666666664) + ((eps * sin(x)) * -0.008333333333333333))) - (sin(x) * -0.16666666666666666))))) - sin(x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.008333333333333333 \cdot \left(\varepsilon \cdot \sin x\right) + 0.041666666666666664 \cdot \cos x\right) - -0.16666666666666666 \cdot \sin x\right)\right) - \sin x\right)} \]

4. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot 0.041666666666666664 + \left(\varepsilon \cdot \sin x\right) \cdot -0.008333333333333333\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \cos x\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (*
    eps
    (+
     (* -0.5 (cos x))
     (*
      eps
      (-
       (* 0.041666666666666664 (* eps (cos x)))
       (* (sin x) -0.16666666666666666)))))
   (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (eps * ((0.041666666666666664 * (eps * cos(x))) - (sin(x) * -0.16666666666666666))))) - sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (((-0.5d0) * cos(x)) + (eps * ((0.041666666666666664d0 * (eps * cos(x))) - (sin(x) * (-0.16666666666666666d0)))))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * Math.cos(x)) + (eps * ((0.041666666666666664 * (eps * Math.cos(x))) - (Math.sin(x) * -0.16666666666666666))))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * ((-0.5 * math.cos(x)) + (eps * ((0.041666666666666664 * (eps * math.cos(x))) - (math.sin(x) * -0.16666666666666666))))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(Float64(-0.5 * cos(x)) + Float64(eps * Float64(Float64(0.041666666666666664 * Float64(eps * cos(x))) - Float64(sin(x) * -0.16666666666666666))))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * ((-0.5 * cos(x)) + (eps * ((0.041666666666666664 * (eps * cos(x))) - (sin(x) * -0.16666666666666666))))) - sin(x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(0.041666666666666664 * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \cos x\right) - -0.16666666666666666 \cdot \sin x\right)\right) - \sin x\right)} \]

4. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \varepsilon \cdot \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \cos x\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. diff-cos 80.8%

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

   2. div-inv 80.8%

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

   3. associate--l+ 80.8%

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

   4. metadata-eval 80.8%

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

   5. div-inv 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   6. +-commutative 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(\color{blue}{\left(\varepsilon + x\right)} + x\right) \cdot \frac{1}{2}\right)\right) \]

   7. associate-+l+ 80.8%

      \[\leadsto -2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\color{blue}{\left(\varepsilon + \left(x + x\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   8. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\left(x + \left(\varepsilon - x\right)\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r- 80.8%

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

   2. +-commutative 80.8%

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

   3. associate--l+ 99.5%

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

   4. +-inverses 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(\varepsilon + \color{blue}{0}\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   5. +-commutative 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(0 + \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   6. *-lft-identity 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(0 + \color{blue}{1 \cdot \varepsilon}\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   7. metadata-eval 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\left(0 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   8. cancel-sign-sub-inv 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   9. neg-sub0 99.5%

      \[\leadsto -2 \cdot \left(\sin \left(\color{blue}{\left(--1 \cdot \varepsilon\right)} \cdot 0.5\right) \cdot \sin \left(\left(\varepsilon + \left(x + x\right)\right) \cdot 0.5\right)\right) \]

   10. mul-1-neg 99.5%

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

   11. remove-double-neg 99.5%

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

   12. *-commutative 99.5%

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

   13. +-commutative 99.5%

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

   14. count-2 99.5%

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

   15. fma-define 99.5%

       \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)}\right)\right) \]

6. Simplified 99.5%

   \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)} \]

7. Taylor expanded in x around -inf 99.5%

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

8. Final simplification 99.5%

   \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right) \]
9. Add Preprocessing

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot \left(0.16666666666666666 + x \cdot \left(\varepsilon \cdot -0.020833333333333332 + x \cdot -0.008333333333333333\right)\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 8.5e-9)
   (*
    eps
    (+
     (* eps -0.5)
     (*
      x
      (+
       (*
        x
        (+
         (* eps 0.25)
         (*
          x
          (+
           0.16666666666666666
           (*
            x
            (+ (* eps -0.020833333333333332) (* x -0.008333333333333333)))))))
       -1.0))))
   (- (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 8.5e-9) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
	} else {
		tmp = cos((eps + x)) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 8.5d-9) then
        tmp = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * (0.16666666666666666d0 + (x * ((eps * (-0.020833333333333332d0)) + (x * (-0.008333333333333333d0)))))))) + (-1.0d0))))
    else
        tmp = cos((eps + x)) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 8.5e-9) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
	} else {
		tmp = Math.cos((eps + x)) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 8.5e-9:
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)))
	else:
		tmp = math.cos((eps + x)) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 8.5e-9)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(Float64(eps * -0.020833333333333332) + Float64(x * -0.008333333333333333))))))) + -1.0))));
	else
		tmp = Float64(cos(Float64(eps + x)) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 8.5e-9)
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
	else
		tmp = cos((eps + x)) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 8.5e-9], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * N[(0.16666666666666666 + N[(x * N[(N[(eps * -0.020833333333333332), $MachinePrecision] + N[(x * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.5e-9

   1. Initial program 53.6%

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

   3. Taylor expanded in eps around 0 99.8%

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

      1. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 98.3%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.25 \cdot \varepsilon + x \cdot \left(0.16666666666666666 + x \cdot \left(-0.020833333333333332 \cdot \varepsilon + -0.008333333333333333 \cdot x\right)\right)\right) - 1\right)\right)} \]

   if 8.5e-9 < eps

   1. Initial program 67.5%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot \left(0.16666666666666666 + x \cdot \left(\varepsilon \cdot -0.020833333333333332 + x \cdot -0.008333333333333333\right)\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

   \[\leadsto \varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x\right) \]
7. Add Preprocessing

Alternative 8: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2.25e-14)
   (*
    eps
    (+
     (* eps -0.5)
     (* x (+ (* x (+ (* eps 0.25) (* x 0.16666666666666666))) -1.0))))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2.25e-14) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2.25d-14) then
        tmp = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2.25e-14) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2.25e-14:
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)))
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2.25e-14)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * 0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2.25e-14)
		tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2.25e-14], N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.2499999999999999e-14

   1. Initial program 54.1%

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

   3. Taylor expanded in eps around 0 99.8%

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

      1. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

   if 2.2499999999999999e-14 < eps

   1. Initial program 53.6%

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

   3. Taylor expanded in eps around 0 53.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 53.2%

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

      2. mul-1-neg 53.2%

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

   5. Simplified 53.2%

      \[\leadsto \color{blue}{\left(-\varepsilon\right) \cdot \sin x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot \left(0.16666666666666666 + x \cdot \left(\varepsilon \cdot -0.020833333333333332 + x \cdot -0.008333333333333333\right)\right)\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (*
    x
    (+
     (*
      x
      (+
       (* eps 0.25)
       (*
        x
        (+
         0.16666666666666666
         (*
          x
          (+ (* eps -0.020833333333333332) (* x -0.008333333333333333)))))))
     -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * (0.16666666666666666d0 + (x * ((eps * (-0.020833333333333332d0)) + (x * (-0.008333333333333333d0)))))))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(Float64(eps * -0.020833333333333332) + Float64(x * -0.008333333333333333))))))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * (0.16666666666666666 + (x * ((eps * -0.020833333333333332) + (x * -0.008333333333333333))))))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * N[(0.16666666666666666 + N[(x * N[(N[(eps * -0.020833333333333332), $MachinePrecision] + N[(x * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.25 \cdot \varepsilon + x \cdot \left(0.16666666666666666 + x \cdot \left(-0.020833333333333332 \cdot \varepsilon + -0.008333333333333333 \cdot x\right)\right)\right) - 1\right)\right)} \]

7. Final simplification 95.3%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot \left(0.16666666666666666 + x \cdot \left(\varepsilon \cdot -0.020833333333333332 + x \cdot -0.008333333333333333\right)\right)\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 10: 98.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (* x (+ (* x (+ (* eps 0.25) (* x 0.16666666666666666))) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(eps * 0.25) + Float64(x * 0.16666666666666666))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * ((eps * 0.25) + (x * 0.16666666666666666))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(eps * 0.25), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.2%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Final simplification 95.2%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 11: 97.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* 0.25 (* eps x)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((0.25d0 * (eps * x)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.25 * Float64(eps * x)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.25 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.1%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]

7. Final simplification 95.1%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 12: 97.5% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * ((eps * -0.5) - x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.1%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-1 \cdot x + -0.5 \cdot \varepsilon\right)} \]
7. Step-by-step derivation

   1. +-commutative 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. *-commutative 95.1%

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

8. Simplified 95.1%

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

9. Final simplification 95.1%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 - x\right) \]
10. Add Preprocessing

Alternative 13: 78.5% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * -x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 79.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
4. Step-by-step derivation

   1. associate-*r* 79.2%

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

   2. mul-1-neg 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in x around 0 76.9%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)} \]
7. Step-by-step derivation

   1. associate-*r* 76.9%

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

   2. neg-mul-1 76.9%

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

8. Simplified 76.9%

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

9. Final simplification 76.9%

   \[\leadsto \varepsilon \cdot \left(-x\right) \]
10. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((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) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024053 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

  (- (cos (+ x eps)) (cos x)))