2cos (problem 3.3.5)

Percentage Accurate: 52.5% → 99.7%

Time: 16.1s

Alternatives: 8

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (sin (+ (* 0.5 eps) x)) (* -2.0 (expm1 (log1p (sin (* 0.5 eps)))))))

C

double code(double x, double eps) {
	return sin(((0.5 * eps) + x)) * (-2.0 * expm1(log1p(sin((0.5 * eps)))));
}

Java

public static double code(double x, double eps) {
	return Math.sin(((0.5 * eps) + x)) * (-2.0 * Math.expm1(Math.log1p(Math.sin((0.5 * eps)))));
}

Python

def code(x, eps):
	return math.sin(((0.5 * eps) + x)) * (-2.0 * math.expm1(math.log1p(math.sin((0.5 * eps)))))

Julia

function code(x, eps)
	return Float64(sin(Float64(Float64(0.5 * eps) + x)) * Float64(-2.0 * expm1(log1p(sin(Float64(0.5 * eps))))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(N[(0.5 * eps), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[(Exp[N[Log[1 + N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. diff-cos 79.9%

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

   2. div-inv 79.9%

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

   3. associate--l+ 79.9%

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

   4. metadata-eval 79.9%

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

   5. div-inv 79.9%

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

   6. +-commutative 79.9%

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

   7. associate-+l+ 79.9%

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

   8. metadata-eval 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*r* 79.9%

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

   2. *-commutative 79.9%

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

   3. *-commutative 79.9%

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

   4. +-commutative 79.9%

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

   5. count-2 79.9%

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

   6. fma-define 79.9%

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

   7. *-commutative 79.9%

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

   8. associate-+r- 79.9%

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

   9. +-commutative 79.9%

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

   10. associate--l+ 99.6%

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

   11. +-inverses 99.6%

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

   12. distribute-lft-in 99.6%

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

   13. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 99.6%

   \[\leadsto \sin \color{blue}{\left(x + 0.5 \cdot \varepsilon\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 99.6%

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

9. Simplified 99.6%

   \[\leadsto \sin \color{blue}{\left(0.5 \cdot \varepsilon + x\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right) \]
10. Step-by-step derivation

    1. expm1-log1p-u 99.7%

       \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon + 0\right)\right)\right)}\right) \]

    2. expm1-undefine 52.3%

       \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon + 0\right)\right)} - 1\right)}\right) \]

    3. +-rgt-identity 52.3%

       \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right)} - 1\right)\right) \]

11. Applied egg-rr 52.3%

    \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon\right)\right)} - 1\right)}\right) \]
12. Step-by-step derivation

    1. expm1-define 99.7%

       \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon\right)\right)\right)}\right) \]

13. Simplified 99.7%

    \[\leadsto \sin \left(0.5 \cdot \varepsilon + x\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
14. Add Preprocessing

Alternative 2: 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 51.6%

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

   1. diff-cos 79.9%

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

   2. div-inv 79.9%

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

   3. associate--l+ 79.9%

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

   4. metadata-eval 79.9%

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

   5. div-inv 79.9%

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

   6. +-commutative 79.9%

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

   7. associate-+l+ 79.9%

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

   8. metadata-eval 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*r* 79.9%

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

   2. *-commutative 79.9%

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

   3. *-commutative 79.9%

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

   4. +-commutative 79.9%

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

   5. count-2 79.9%

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

   6. fma-define 79.9%

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

   7. *-commutative 79.9%

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

   8. associate-+r- 79.9%

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

   9. +-commutative 79.9%

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

   10. associate--l+ 99.6%

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

   11. +-inverses 99.6%

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

   12. distribute-lft-in 99.6%

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

   13. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 99.6%

   \[\leadsto \sin \color{blue}{\left(x + 0.5 \cdot \varepsilon\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 99.6%

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

9. Simplified 99.6%

   \[\leadsto \sin \color{blue}{\left(0.5 \cdot \varepsilon + x\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right) \]
10. Step-by-step derivation

    1. +-rgt-identity 99.6%

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

    2. *-commutative 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. diff-cos 79.9%

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

   2. div-inv 79.9%

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

   3. associate--l+ 79.9%

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

   4. metadata-eval 79.9%

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

   5. div-inv 79.9%

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

   6. +-commutative 79.9%

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

   7. associate-+l+ 79.9%

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

   8. metadata-eval 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*r* 79.9%

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

   2. *-commutative 79.9%

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

   3. *-commutative 79.9%

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

   4. +-commutative 79.9%

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

   5. count-2 79.9%

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

   6. fma-define 79.9%

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

   7. *-commutative 79.9%

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

   8. associate-+r- 79.9%

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

   9. +-commutative 79.9%

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

   10. associate--l+ 99.6%

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

   11. +-inverses 99.6%

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

   12. distribute-lft-in 99.6%

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

   13. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 99.6%

   \[\leadsto \sin \color{blue}{\left(x + 0.5 \cdot \varepsilon\right)} \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon + 0\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in eps around 0 99.1%

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

    1. mul-1-neg 99.1%

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

12. Simplified 99.1%

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

Alternative 5: 98.9% 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 51.6%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

8. Simplified 98.5%

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

Alternative 6: 97.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 96.6%

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

7. Taylor expanded in eps around 0 96.7%

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

   1. +-commutative 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. *-commutative 96.7%

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

9. Simplified 96.7%

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

Alternative 7: 78.9% 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 51.6%

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

3. Taylor expanded in eps around 0 79.3%

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

   1. associate-*r* 79.3%

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

   2. mul-1-neg 79.3%

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

5. Simplified 79.3%

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

6. Taylor expanded in x around 0 77.9%

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

   1. associate-*r* 77.9%

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

   2. mul-1-neg 77.9%

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

8. Simplified 77.9%

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

9. Final simplification 77.9%

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

Alternative 8: 51.2% 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 51.6%

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

3. Taylor expanded in x around 0 49.8%

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

4. Taylor expanded in eps around 0 49.7%

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

   1. metadata-eval 49.7%

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

6. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{0} \]
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 2024139 
(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)))