2cos (problem 3.3.5)

Percentage Accurate: 52.8% → 99.7%

Time: 18.8s

Alternatives: 6

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: 52.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[Sin[N[(N[(eps * 0.5), $MachinePrecision] + x), $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 81.6%

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

   2. div-inv 81.6%

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

   3. associate--l+ 81.6%

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

   4. metadata-eval 81.6%

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

   5. div-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. associate-+l+ 81.6%

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

   8. metadata-eval 81.6%

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

4. Applied egg-rr 81.6%

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

   1. associate-*r* 81.6%

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

   2. *-commutative 81.6%

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

   3. associate-*l* 81.6%

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

   4. +-commutative 81.6%

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

   5. associate-+l- 99.8%

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

   6. +-inverses 99.8%

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

   7. --rgt-identity 99.8%

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

   8. remove-double-neg 99.8%

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

   9. mul-1-neg 99.8%

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

   10. mul-1-neg 99.8%

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

   11. remove-double-neg 99.8%

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

   12. *-commutative 99.8%

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

   13. +-commutative 99.8%

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

   14. distribute-lft-in 99.8%

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

   15. count-2 99.8%

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

   16. associate-*r* 99.8%

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

   17. metadata-eval 99.8%

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

   18. *-lft-identity 99.8%

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

   19. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

   4. *-commutative 99.6%

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

5. Simplified 99.6%

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

   1. add-exp-log 99.0%

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

7. Applied egg-rr 99.0%

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

   1. rem-exp-log 99.6%

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

   2. unpow2 99.6%

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

   3. associate-*r* 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*r* 99.6%

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

    2. *-commutative 99.6%

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

    3. distribute-rgt-out-- 99.6%

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

    4. *-commutative 99.6%

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

    5. associate-*r* 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

   4. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.5%

   \[\leadsto -0.5 \cdot \color{blue}{{\varepsilon}^{2}} - \varepsilon \cdot \sin x \]
7. Step-by-step derivation

   1. unpow2 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

   4. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.5%

   \[\leadsto -0.5 \cdot \color{blue}{{\varepsilon}^{2}} - \varepsilon \cdot \sin x \]
7. Step-by-step derivation

   1. unpow2 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in eps around 0 99.5%

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

    1. mul-1-neg 99.5%

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

    2. +-commutative 99.5%

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

    3. *-commutative 99.5%

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

    4. distribute-lft-neg-in 99.5%

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

    5. unpow2 99.5%

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

    6. associate-*r* 99.5%

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

    7. distribute-rgt-in 99.5%

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

    8. sub-neg 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 5: 97.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - N[(eps * 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 99.6%

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

   4. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0 99.5%

   \[\leadsto -0.5 \cdot \color{blue}{{\varepsilon}^{2}} - \varepsilon \cdot \sin x \]
7. Step-by-step derivation

   1. unpow2 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around 0 98.8%

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

10. Final simplification 98.8%

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

Alternative 6: 79.1% 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 80.8%

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

   1. associate-*r* 80.8%

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

   2. mul-1-neg 80.8%

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

5. Simplified 80.8%

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

6. Taylor expanded in x around 0 80.2%

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

   1. associate-*r* 80.2%

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

   2. mul-1-neg 80.2%

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

8. Simplified 80.2%

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

9. Final simplification 80.2%

   \[\leadsto \varepsilon \cdot \left(-x\right) \]
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 2024046 
(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)))