2cos (problem 3.3.5)

Percentage Accurate: 52.7% → 99.8%

Time: 17.1s

Alternatives: 7

Speedup: 2.0×

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: 52.7% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \left(-2 \cdot t\_0\right) \cdot \left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \sin x + t\_0 \cdot \cos x\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* (* -2.0 t_0) (+ (* (cos (* eps 0.5)) (sin x)) (* t_0 (cos x))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return (-2.0 * t_0) * ((cos((eps * 0.5)) * sin(x)) + (t_0 * cos(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = ((-2.0d0) * t_0) * ((cos((eps * 0.5d0)) * sin(x)) + (t_0 * cos(x)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	return (-2.0 * t_0) * ((Math.cos((eps * 0.5)) * Math.sin(x)) + (t_0 * Math.cos(x)));
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return (-2.0 * t_0) * ((math.cos((eps * 0.5)) * math.sin(x)) + (t_0 * math.cos(x)))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(Float64(-2.0 * t_0) * Float64(Float64(cos(Float64(eps * 0.5)) * sin(x)) + Float64(t_0 * cos(x))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = (-2.0 * t_0) * ((cos((eps * 0.5)) * sin(x)) + (t_0 * cos(x)));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(-2.0 * t$95$0), $MachinePrecision] * N[(N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.3%

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

   1. diff-cos 81.5%

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

   2. div-inv 81.5%

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

   3. associate--l+ 81.5%

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

   4. metadata-eval 81.5%

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

   5. div-inv 81.5%

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

   6. +-commutative 81.5%

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

   7. associate-+l+ 81.5%

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

   8. metadata-eval 81.5%

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

4. Applied egg-rr 81.5%

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

   1. associate-*r* 81.5%

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

   2. *-commutative 81.5%

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

   3. *-commutative 81.5%

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

   4. +-commutative 81.5%

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

   5. count-2 81.5%

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

   6. fma-define 81.5%

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

   7. *-commutative 81.5%

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

   8. associate-+r- 81.5%

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

   9. +-commutative 81.5%

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

   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. *-commutative 99.7%

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

    2. +-rgt-identity 99.7%

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

    3. *-commutative 99.7%

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

    4. sin-sum 99.8%

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

    5. distribute-lft-in 99.7%

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

    6. *-commutative 99.7%

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

11. Applied egg-rr 99.7%

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

    1. distribute-lft-out 99.8%

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

    2. *-commutative 99.8%

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

    3. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5 + \varepsilon \cdot \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \cos x\right) - \sin x \cdot -0.16666666666666666\right)\right) - \sin x\right) \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. diff-cos 81.5%

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

   2. div-inv 81.5%

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

   3. associate--l+ 81.5%

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

   4. metadata-eval 81.5%

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

   5. div-inv 81.5%

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

   6. +-commutative 81.5%

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

   7. associate-+l+ 81.5%

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

   8. metadata-eval 81.5%

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

4. Applied egg-rr 81.5%

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

   1. associate-*r* 81.5%

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

   2. *-commutative 81.5%

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

   3. *-commutative 81.5%

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

   4. +-commutative 81.5%

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

   5. count-2 81.5%

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

   6. fma-define 81.5%

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

   7. *-commutative 81.5%

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

   8. associate-+r- 81.5%

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

   9. +-commutative 81.5%

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

   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 inf 99.7%

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

    1. metadata-eval 99.7%

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

    2. distribute-lft-neg-in 99.7%

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

    3. distribute-lft-neg-in 99.7%

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

    4. metadata-eval 99.7%

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

    5. *-commutative 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 4: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. diff-cos 81.5%

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

   2. div-inv 81.5%

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

   3. associate--l+ 81.5%

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

   4. metadata-eval 81.5%

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

   5. div-inv 81.5%

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

   6. +-commutative 81.5%

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

   7. associate-+l+ 81.5%

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

   8. metadata-eval 81.5%

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

4. Applied egg-rr 81.5%

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

   1. associate-*r* 81.5%

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

   2. *-commutative 81.5%

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

   3. *-commutative 81.5%

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

   4. +-commutative 81.5%

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

   5. count-2 81.5%

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

   6. fma-define 81.5%

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

   7. *-commutative 81.5%

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

   8. associate-+r- 81.5%

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

   9. +-commutative 81.5%

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

   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. mul-1-neg 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. Final simplification 99.3%

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

Alternative 5: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. diff-cos 81.5%

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

   2. *-commutative 81.5%

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

   3. div-inv 81.5%

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

   4. associate--l+ 81.5%

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

   5. metadata-eval 81.5%

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

   6. div-inv 81.5%

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

   7. +-commutative 81.5%

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

   8. associate-+l+ 81.5%

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

   9. metadata-eval 81.5%

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in eps around 0 79.5%

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

   1. associate-*r* 79.5%

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

   2. *-commutative 79.5%

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

7. Simplified 79.5%

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

8. Final simplification 79.5%

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

Alternative 6: 80.2% 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 53.3%

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

3. Taylor expanded in eps around 0 79.5%

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

   1. associate-*r* 79.5%

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

   2. mul-1-neg 79.5%

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

5. Simplified 79.5%

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

6. Final simplification 79.5%

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

Alternative 7: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos \varepsilon + -1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ (cos eps) -1.0))

C

double code(double x, double eps) {
	return cos(eps) + -1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos(eps) + (-1.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.cos(eps) + -1.0;
}

Python

def code(x, eps):
	return math.cos(eps) + -1.0

Julia

function code(x, eps)
	return Float64(cos(eps) + -1.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = cos(eps) + -1.0;
end

Wolfram

code[x_, eps_] := N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0 52.0%

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

4. Final simplification 52.0%

   \[\leadsto \cos \varepsilon + -1 \]
5. 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 2024113 
(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)))