2cos (problem 3.3.5)

Percentage Accurate: 52.3% → 99.2%

Time: 19.0s

Alternatives: 9

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: 52.3% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (x * (eps * 0.16666666666666666)))) - 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)) + (x * (eps * 0.16666666666666666d0)))) - sin(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 99.8%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \color{blue}{0.16666666666666666 \cdot \left(\varepsilon \cdot x\right)}\right) - \sin x\right) \]
5. Step-by-step derivation

   1. associate-*r* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.6%

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

   1. diff-cos 77.7%

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

   2. *-commutative 77.7%

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

   3. div-inv 77.7%

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

   4. associate--l+ 77.7%

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

   5. metadata-eval 77.7%

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

   6. div-inv 77.7%

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

   7. +-commutative 77.7%

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

   8. associate-+l+ 77.7%

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

   9. metadata-eval 77.7%

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

4. Applied egg-rr 77.7%

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

5. Taylor expanded in x around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.2% 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 48.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(\left(0.16666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right)\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (*
    x
    (+
     (+
      (* 0.16666666666666666 (pow eps 2.0))
      (* x (+ (* x 0.16666666666666666) (* eps 0.25))))
     -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * (((0.16666666666666666 * pow(eps, 2.0)) + (x * ((x * 0.16666666666666666) + (eps * 0.25)))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * (((0.16666666666666666d0 * (eps ** 2.0d0)) + (x * ((x * 0.16666666666666666d0) + (eps * 0.25d0)))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * (((0.16666666666666666 * Math.pow(eps, 2.0)) + (x * ((x * 0.16666666666666666) + (eps * 0.25)))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * (((0.16666666666666666 * math.pow(eps, 2.0)) + (x * ((x * 0.16666666666666666) + (eps * 0.25)))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(Float64(0.16666666666666666 * (eps ^ 2.0)) + Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(eps * 0.25)))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * (((0.16666666666666666 * (eps ^ 2.0)) + (x * ((x * 0.16666666666666666) + (eps * 0.25)))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(N[(0.16666666666666666 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(eps * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 99.8%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x + \color{blue}{0.16666666666666666 \cdot \left(\varepsilon \cdot x\right)}\right) - \sin x\right) \]
5. Step-by-step derivation

   1. associate-*r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 98.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(\left(0.16666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right)\right) - 1\right)\right)} \]

8. Final simplification 98.5%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(\left(0.16666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right)\right) + -1\right)\right) \]
9. Add Preprocessing

Alternative 5: 98.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (* x (+ (* x (+ (* x 0.16666666666666666) (* eps 0.25))) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * ((x * 0.16666666666666666d0) + (eps * 0.25d0))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(eps * 0.25))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(eps * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 98.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Final simplification 98.5%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 6: 97.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* 0.25 (* eps x)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((0.25d0 * (eps * x)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.25 * Float64(eps * x)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.25 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 97.6%

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

7. Final simplification 97.6%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 7: 97.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + \left(x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25\right)\right) - x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (- (* x (* x (* eps 0.25))) x))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + ((x * (x * (eps * 0.25))) - x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + ((x * (x * (eps * 0.25d0))) - x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + ((x * (x * (eps * 0.25))) - x));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + ((x * (x * (eps * 0.25))) - x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(Float64(x * Float64(x * Float64(eps * 0.25))) - x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + ((x * (x * (eps * 0.25))) - x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(N[(x * N[(x * N[(eps * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 97.6%

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

   1. sub-neg 97.6%

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

   2. distribute-rgt-in 97.6%

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

   3. associate-*r* 97.6%

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

   4. metadata-eval 97.6%

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

8. Applied egg-rr 97.6%

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

9. Final simplification 97.6%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + \left(x \cdot \left(x \cdot \left(\varepsilon \cdot 0.25\right)\right) - x\right)\right) \]
10. Add Preprocessing

Alternative 8: 97.5% 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 48.6%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 97.5%

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

   1. mul-1-neg 97.5%

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

   2. +-commutative 97.5%

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

   3. unsub-neg 97.5%

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

   4. *-commutative 97.5%

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 9: 78.4% 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 48.6%

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

3. Taylor expanded in eps around 0 80.0%

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

   1. mul-1-neg 80.0%

      \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

   2. *-commutative 80.0%

      \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

   3. distribute-rgt-neg-in 80.0%

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

5. Simplified 80.0%

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

6. Taylor expanded in x around 0 78.8%

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

   1. associate-*r* 78.8%

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

   2. mul-1-neg 78.8%

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

8. Simplified 78.8%

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

9. Final simplification 78.8%

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

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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