2cos (problem 3.3.5)

Percentage Accurate: 52.5% → 99.7%

Time: 21.9s

Alternatives: 7

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: 52.5% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma 0.16666666666666666 (pow eps 3.0) (- eps))
  (sin x)
  (*
   (cos x)
   (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (pow eps 2.0))))))

C

double code(double x, double eps) {
	return fma(fma(0.16666666666666666, pow(eps, 3.0), -eps), sin(x), (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * pow(eps, 2.0)))));
}

Julia

function code(x, eps)
	return fma(fma(0.16666666666666666, (eps ^ 3.0), Float64(-eps)), sin(x), Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * (eps ^ 2.0)))))
end

Wolfram

code[x_, eps_] := N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision] + (-eps)), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]

   2. associate-+r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} + -1 \cdot \left(\varepsilon \cdot \sin x\right) \]

   3. associate-+l+ 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

   4. associate-*r* 99.7%

      \[\leadsto \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   5. associate-*r* 99.7%

      \[\leadsto \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   6. distribute-rgt-out 99.7%

      \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   7. associate-*r* 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x} + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   8. associate-*r* 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

   9. distribute-rgt-out 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + -1 \cdot \varepsilon\right)} \]

   10. mul-1-neg 99.7%

       \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \color{blue}{\left(-\varepsilon\right)}\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right) \cdot \sin x} + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) \]

   3. fma-def 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right), \sin x, \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right)} \]

   4. fma-def 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right)}, \sin x, \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) \]

   5. +-commutative 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)}\right) \]

   6. fma-def 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)}\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)}\right) \]

9. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)}\right) \]

10. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, {\varepsilon}^{3}, -\varepsilon\right), \sin x, \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\right) \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \sin x \cdot \left({\varepsilon}^{2} \cdot -0.125 + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* -2.0 (sin (* eps 0.5)))
  (+
   (* (cos x) (+ (* eps 0.5) (* (pow eps 3.0) -0.020833333333333332)))
   (* (sin x) (+ (* (pow eps 2.0) -0.125) 1.0)))))

C

double code(double x, double eps) {
	return (-2.0 * sin((eps * 0.5))) * ((cos(x) * ((eps * 0.5) + (pow(eps, 3.0) * -0.020833333333333332))) + (sin(x) * ((pow(eps, 2.0) * -0.125) + 1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((-2.0d0) * sin((eps * 0.5d0))) * ((cos(x) * ((eps * 0.5d0) + ((eps ** 3.0d0) * (-0.020833333333333332d0)))) + (sin(x) * (((eps ** 2.0d0) * (-0.125d0)) + 1.0d0)))
end function

Java

public static double code(double x, double eps) {
	return (-2.0 * Math.sin((eps * 0.5))) * ((Math.cos(x) * ((eps * 0.5) + (Math.pow(eps, 3.0) * -0.020833333333333332))) + (Math.sin(x) * ((Math.pow(eps, 2.0) * -0.125) + 1.0)));
}

Python

def code(x, eps):
	return (-2.0 * math.sin((eps * 0.5))) * ((math.cos(x) * ((eps * 0.5) + (math.pow(eps, 3.0) * -0.020833333333333332))) + (math.sin(x) * ((math.pow(eps, 2.0) * -0.125) + 1.0)))

Julia

function code(x, eps)
	return Float64(Float64(-2.0 * sin(Float64(eps * 0.5))) * Float64(Float64(cos(x) * Float64(Float64(eps * 0.5) + Float64((eps ^ 3.0) * -0.020833333333333332))) + Float64(sin(x) * Float64(Float64((eps ^ 2.0) * -0.125) + 1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (-2.0 * sin((eps * 0.5))) * ((cos(x) * ((eps * 0.5) + ((eps ^ 3.0) * -0.020833333333333332))) + (sin(x) * (((eps ^ 2.0) * -0.125) + 1.0)));
end

Wolfram

code[x_, eps_] := N[(N[(-2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * 0.5), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 2.0], $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. diff-cos 79.4%

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

   2. associate-*r* 79.4%

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

   3. div-inv 79.4%

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

   4. associate--l+ 79.4%

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

   5. metadata-eval 79.4%

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

   6. div-inv 79.4%

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

   7. +-commutative 79.4%

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

   8. associate-+l+ 79.4%

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

   9. metadata-eval 79.4%

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in x around 0 99.7%

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

6. Taylor expanded in eps around 0 99.8%

   \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-+r+ 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]

   2. +-commutative 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \color{blue}{\left(\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)} \]

   3. +-commutative 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   4. associate-*r* 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   5. associate-*r* 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   6. distribute-rgt-out 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   7. *-commutative 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\color{blue}{\varepsilon \cdot 0.5} + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   8. *-commutative 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + \color{blue}{{\varepsilon}^{3} \cdot -0.020833333333333332}\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right) \]

   9. associate-*r* 99.8%

      \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right)\right) \]

   10. distribute-rgt1-in 99.8%

       \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x}\right) \]

   11. *-commutative 99.8%

       \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \left(\color{blue}{{\varepsilon}^{2} \cdot -0.125} + 1\right) \cdot \sin x\right) \]

8. Simplified 99.8%

   \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \color{blue}{\left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \left({\varepsilon}^{2} \cdot -0.125 + 1\right) \cdot \sin x\right)} \]

9. Final simplification 99.8%

   \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \left(\varepsilon \cdot 0.5 + {\varepsilon}^{3} \cdot -0.020833333333333332\right) + \sin x \cdot \left({\varepsilon}^{2} \cdot -0.125 + 1\right)\right) \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* (cos x) (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (pow eps 2.0))))
  (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))

C

double code(double x, double eps) {
	return (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * pow(eps, 2.0)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps ** 2.0d0)))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
end function

Java

public static double code(double x, double eps) {
	return (Math.cos(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * Math.pow(eps, 2.0)))) + (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps));
}

Python

def code(x, eps):
	return (math.cos(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * math.pow(eps, 2.0)))) + (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps))

Julia

function code(x, eps)
	return Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * (eps ^ 2.0)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps ^ 2.0)))) + (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) - eps));
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]

   2. associate-+r+ 99.7%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} + -1 \cdot \left(\varepsilon \cdot \sin x\right) \]

   3. associate-+l+ 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

   4. associate-*r* 99.7%

      \[\leadsto \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   5. associate-*r* 99.7%

      \[\leadsto \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   6. distribute-rgt-out 99.7%

      \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   7. associate-*r* 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x} + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

   8. associate-*r* 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

   9. distribute-rgt-out 99.7%

      \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + -1 \cdot \varepsilon\right)} \]

   10. mul-1-neg 99.7%

       \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \color{blue}{\left(-\varepsilon\right)}\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)} \]

6. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)} \]

7. Final simplification 99.7%

   \[\leadsto \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right) \]
8. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((eps * 0.5))) * sin((0.5 * (eps + (x + 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((0.5d0 * (eps + (x + x))))
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 + x))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. diff-cos 79.4%

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

   2. associate-*r* 79.4%

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

   3. div-inv 79.4%

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

   4. associate--l+ 79.4%

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

   5. metadata-eval 79.4%

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

   6. div-inv 79.4%

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

   7. +-commutative 79.4%

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

   8. associate-+l+ 79.4%

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

   9. metadata-eval 79.4%

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in x around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. mul-1-neg 99.2%

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

   3. unsub-neg 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 97.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right) + -0.5 \cdot {\varepsilon}^{2}} \]
7. Step-by-step derivation

   1. +-commutative 97.3%

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

   2. mul-1-neg 97.3%

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

   3. unsub-neg 97.3%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x} \]

8. Simplified 97.3%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x} \]

9. Final simplification 97.3%

   \[\leadsto -0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x \]
10. Add Preprocessing

Alternative 6: 79.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in eps around 0 78.7%

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

   1. mul-1-neg 78.7%

      \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

   2. *-commutative 78.7%

      \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

   3. distribute-rgt-neg-in 78.7%

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

5. Simplified 78.7%

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

6. Final simplification 78.7%

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

Alternative 7: 78.8% 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 50.0%

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

3. Taylor expanded in eps around 0 78.7%

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

   1. mul-1-neg 78.7%

      \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

   2. *-commutative 78.7%

      \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

   3. distribute-rgt-neg-in 78.7%

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

5. Simplified 78.7%

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

6. Taylor expanded in x around 0 77.6%

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

   1. mul-1-neg 77.6%

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

   2. distribute-rgt-neg-in 77.6%

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

8. Simplified 77.6%

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

9. Final simplification 77.6%

   \[\leadsto \varepsilon \cdot \left(-x\right) \]
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 2024029 
(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)))

  :herbie-target
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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