2cos (problem 3.3.5)

Percentage Accurate: 53.0% → 99.7%

Time: 15.2s

Alternatives: 5

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: 53.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. diff-cos 77.9%

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

   2. div-inv 77.9%

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

   3. associate--l+ 77.9%

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

   4. metadata-eval 77.9%

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

   5. div-inv 77.9%

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

   6. +-commutative 77.9%

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

   7. associate-+l+ 77.9%

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

   8. metadata-eval 77.9%

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

4. Applied egg-rr 77.9%

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

   1. associate-*r* 77.9%

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

   2. *-commutative 77.9%

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

   3. *-commutative 77.9%

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

   4. +-commutative 77.9%

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

   5. count-2 77.9%

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

   6. fma-define 77.9%

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

   7. *-commutative 77.9%

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

   8. associate-+r- 77.9%

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

   9. +-commutative 77.9%

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

   10. associate--l+ 99.7%

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

   11. +-inverses 99.7%

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

   12. distribute-lft-in 99.7%

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

   13. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

    1. +-rgt-identity 99.7%

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

    2. *-commutative 99.7%

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

11. Applied egg-rr 99.7%

    \[\leadsto \sin \left(x + \varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \]
12. 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 50.1%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. diff-cos 77.9%

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

   2. div-inv 77.9%

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

   3. associate--l+ 77.9%

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

   4. metadata-eval 77.9%

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

   5. div-inv 77.9%

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

   6. +-commutative 77.9%

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

   7. associate-+l+ 77.9%

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

   8. metadata-eval 77.9%

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

4. Applied egg-rr 77.9%

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

   1. associate-*r* 77.9%

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

   2. *-commutative 77.9%

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

   3. *-commutative 77.9%

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

   4. +-commutative 77.9%

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

   5. count-2 77.9%

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

   6. fma-define 77.9%

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

   7. *-commutative 77.9%

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

   8. associate-+r- 77.9%

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

   9. +-commutative 77.9%

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

   10. associate--l+ 99.7%

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

   11. +-inverses 99.7%

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

   12. distribute-lft-in 99.7%

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

   13. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in eps around 0 99.3%

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

    1. neg-mul-1 99.3%

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

12. Simplified 99.3%

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

Alternative 4: 80.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in eps around 0 80.6%

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

   1. associate-*r* 80.6%

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

   2. mul-1-neg 80.6%

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

5. Simplified 80.6%

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

6. Final simplification 80.6%

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

Alternative 5: 79.1% 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 50.1%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2} + x \cdot \left(-1 \cdot \varepsilon + 0.25 \cdot \left({\varepsilon}^{2} \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + 0.25 \cdot \left({\varepsilon}^{2} \cdot x\right)\right) + -0.5 \cdot {\varepsilon}^{2}} \]

   2. fma-define 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \varepsilon + 0.25 \cdot \left({\varepsilon}^{2} \cdot x\right), -0.5 \cdot {\varepsilon}^{2}\right)} \]

   3. +-commutative 97.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.25 \cdot \left({\varepsilon}^{2} \cdot x\right) + -1 \cdot \varepsilon}, -0.5 \cdot {\varepsilon}^{2}\right) \]

   4. mul-1-neg 97.4%

      \[\leadsto \mathsf{fma}\left(x, 0.25 \cdot \left({\varepsilon}^{2} \cdot x\right) + \color{blue}{\left(-\varepsilon\right)}, -0.5 \cdot {\varepsilon}^{2}\right) \]

   5. unsub-neg 97.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.25 \cdot \left({\varepsilon}^{2} \cdot x\right) - \varepsilon}, -0.5 \cdot {\varepsilon}^{2}\right) \]

   6. associate-*r* 97.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(0.25 \cdot {\varepsilon}^{2}\right) \cdot x} - \varepsilon, -0.5 \cdot {\varepsilon}^{2}\right) \]

   7. *-commutative 97.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left({\varepsilon}^{2} \cdot 0.25\right)} \cdot x - \varepsilon, -0.5 \cdot {\varepsilon}^{2}\right) \]

   8. associate-*l* 97.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{{\varepsilon}^{2} \cdot \left(0.25 \cdot x\right)} - \varepsilon, -0.5 \cdot {\varepsilon}^{2}\right) \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, {\varepsilon}^{2} \cdot \left(0.25 \cdot x\right) - \varepsilon, -0.5 \cdot {\varepsilon}^{2}\right)} \]

9. Taylor expanded in eps around 0 79.2%

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

    1. associate-*r* 79.2%

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

    2. neg-mul-1 79.2%

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

11. Simplified 79.2%

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

12. Final simplification 79.2%

    \[\leadsto x \cdot \left(-\varepsilon\right) \]
13. 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 2024111 
(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)))