2cos (problem 3.3.5)

Percentage Accurate: 52.4% → 99.7%

Time: 14.7s

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: 52.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. diff-cos 79.6%

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

   2. div-inv 79.6%

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

   3. associate--l+ 79.6%

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

   4. metadata-eval 79.6%

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

   5. div-inv 79.6%

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

   6. +-commutative 79.6%

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

   7. associate-+l+ 79.7%

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

   8. metadata-eval 79.7%

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

4. Applied egg-rr 79.7%

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

   1. associate-*r* 79.7%

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

   2. *-commutative 79.7%

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

   3. associate-*l* 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-+r- 79.6%

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

   6. +-commutative 79.6%

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

   7. associate--l+ 99.8%

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

   8. +-inverses 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. metadata-eval 99.8%

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

   11. *-commutative 99.8%

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

   12. +-commutative 99.8%

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

   13. count-2 99.8%

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

   14. fma-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around -inf 99.8%

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

8. Taylor expanded in x around inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-in 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-rgt-identity 99.8%

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

10. Simplified 99.8%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. diff-cos 79.6%

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

   2. div-inv 79.6%

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

   3. associate--l+ 79.6%

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

   4. metadata-eval 79.6%

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

   5. div-inv 79.6%

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

   6. +-commutative 79.6%

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

   7. associate-+l+ 79.7%

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

   8. metadata-eval 79.7%

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

4. Applied egg-rr 79.7%

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

   1. associate-*r* 79.7%

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

   2. *-commutative 79.7%

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

   3. associate-*l* 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-+r- 79.6%

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

   6. +-commutative 79.6%

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

   7. associate--l+ 99.8%

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

   8. +-inverses 99.8%

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

   9. distribute-lft-in 99.8%

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

   10. metadata-eval 99.8%

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

   11. *-commutative 99.8%

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

   12. +-commutative 99.8%

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

   13. count-2 99.8%

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

   14. fma-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around -inf 99.8%

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

8. Taylor expanded in x around inf 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. distribute-rgt-in 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-rgt-identity 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in eps around 0 98.9%

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

    1. neg-mul-1 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 3: 79.3% accurate, 2.0× speedup

Math

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

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(Float64(-eps) * 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 51.5%

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

3. Taylor expanded in eps around 0 79.4%

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

   1. associate-*r* 79.4%

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

   2. mul-1-neg 79.4%

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

5. Simplified 79.4%

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

Alternative 4: 78.7% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in eps around 0 79.4%

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

   1. associate-*r* 79.4%

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

   2. mul-1-neg 79.4%

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

5. Simplified 79.4%

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

6. Taylor expanded in x around 0 79.0%

   \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + 0.16666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 79.0%

      \[\leadsto x \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right) + -1 \cdot \varepsilon\right)} \]

   2. mul-1-neg 79.0%

      \[\leadsto x \cdot \left(0.16666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right) + \color{blue}{\left(-\varepsilon\right)}\right) \]

   3. unsub-neg 79.0%

      \[\leadsto x \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right) - \varepsilon\right)} \]

   4. associate-*r* 79.0%

      \[\leadsto x \cdot \left(\color{blue}{\left(0.16666666666666666 \cdot \varepsilon\right) \cdot {x}^{2}} - \varepsilon\right) \]

   5. *-commutative 79.0%

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \left(0.16666666666666666 \cdot \varepsilon\right)} - \varepsilon\right) \]

   6. *-commutative 79.0%

      \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\varepsilon \cdot 0.16666666666666666\right)} - \varepsilon\right) \]

8. Simplified 79.0%

   \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\varepsilon \cdot 0.16666666666666666\right) - \varepsilon\right)} \]
9. Step-by-step derivation

   1. unpow2 79.0%

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

10. Applied egg-rr 79.0%

    \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon \cdot 0.16666666666666666\right) - \varepsilon\right) \]
11. Add Preprocessing

Alternative 5: 78.5% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in eps around 0 79.4%

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

   1. associate-*r* 79.4%

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

   2. mul-1-neg 79.4%

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

5. Simplified 79.4%

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

6. Taylor expanded in x around 0 78.9%

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

7. Final simplification 78.9%

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

Alternative 6: 51.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

4. Taylor expanded in eps around 0 49.5%

   \[\leadsto \color{blue}{1} - 1 \]
5. Step-by-step derivation

   1. metadata-eval 49.5%

      \[\leadsto \color{blue}{0} \]

6. Applied egg-rr 49.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer Target 1: 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 2024157 
(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
  (! :herbie-platform default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

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