2cos (problem 3.3.5)

Percentage Accurate: 52.7% → 99.5%

Time: 20.0s

Alternatives: 8

Speedup: 41.0×

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (eps * ((0.041666666666666664 * (eps * cos(x))) - (-0.16666666666666666 * sin(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)) + (eps * ((0.041666666666666664d0 * (eps * cos(x))) - ((-0.16666666666666666d0) * sin(x)))))) - sin(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 99.0%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.8%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((eps * ((-0.5 * cos(x)) + (eps * (0.041666666666666664 * eps)))) - 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)) + (eps * (0.041666666666666664d0 * eps)))) - sin(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 99.0%

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

4. Taylor expanded in x around 0 98.6%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.5%

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

Alternative 5: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.5%

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

4. Taylor expanded in x around 0 98.1%

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

Alternative 6: 98.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.5%

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

4. Taylor expanded in x around 0 96.9%

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

Alternative 7: 97.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.5%

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

4. Taylor expanded in x around 0 96.6%

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

Alternative 8: 78.7% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* -1.0 (* eps x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in eps around 0 98.5%

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

4. Taylor expanded in x around 0 96.6%

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

5. Taylor expanded in eps around 0 76.5%

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)} \]
6. 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 2024050 -o generate:simplify
(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)))