2cos (problem 3.3.5)

Percentage Accurate: 52.4% → 99.7%

Time: 18.6s

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: 52.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. diff-cos 81.2%

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

   2. div-inv 81.2%

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

   3. associate--l+ 81.2%

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

   4. metadata-eval 81.2%

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

   5. div-inv 81.2%

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

   6. +-commutative 81.2%

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

   7. associate-+l+ 81.2%

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

   8. metadata-eval 81.2%

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

4. Applied egg-rr 81.2%

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

   1. associate-*r* 81.2%

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

   2. *-commutative 81.2%

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

   3. *-commutative 81.2%

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

   4. +-commutative 81.2%

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

   5. count-2 81.2%

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

   6. fma-define 81.2%

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

   7. associate-+r- 81.2%

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

   8. +-commutative 81.2%

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

   9. associate--l+ 99.6%

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

   10. +-inverses 99.6%

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

   11. +-commutative 99.6%

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

   12. *-lft-identity 99.6%

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

   13. metadata-eval 99.6%

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

   14. cancel-sign-sub-inv 99.6%

       \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

   15. neg-sub0 99.6%

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

   16. mul-1-neg 99.6%

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

   17. remove-double-neg 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around -inf 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \left(-2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \]
9. 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 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 97.2%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]
7. Step-by-step derivation

   1. distribute-rgt-in 97.2%

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

   2. *-commutative 97.2%

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

   3. fmm-def 97.2%

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

   4. +-commutative 97.2%

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

   5. *-commutative 97.2%

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

   6. fma-define 97.2%

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

   7. *-commutative 97.2%

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

   8. metadata-eval 97.2%

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

8. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot -0.5\right) \cdot \varepsilon + \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -1\right)\right) \cdot \varepsilon} \]
9. Step-by-step derivation

   1. fma-undefine 97.2%

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

10. Applied egg-rr 97.2%

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

11. Final simplification 97.2%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5\right) + \varepsilon \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25, -1\right)\right) \]
12. Add Preprocessing

Alternative 4: 98.1% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 97.2%

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

7. Taylor expanded in eps around inf 97.2%

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

8. Final simplification 97.2%

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

Alternative 5: 98.1% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 97.2%

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

7. Final simplification 97.2%

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

Alternative 6: 98.0% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 97.2%

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

7. Taylor expanded in x around inf 97.1%

   \[\leadsto \varepsilon \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(0.16666666666666666 \cdot x\right)} - 1\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 7: 97.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 97.2%

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

7. Taylor expanded in x around 0 96.7%

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

   1. associate-*r* 96.7%

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

   2. neg-mul-1 96.7%

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

   3. +-commutative 96.7%

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

   4. distribute-lft-neg-in 96.7%

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

   5. unsub-neg 96.7%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x} \]

9. Simplified 96.7%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2} - \varepsilon \cdot x} \]
10. Step-by-step derivation

    1. unpow2 96.7%

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

11. Applied egg-rr 96.7%

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

12. Final simplification 96.7%

    \[\leadsto -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \varepsilon \cdot x \]
13. 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 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. +-commutative 96.7%

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

   3. unsub-neg 96.7%

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

   4. *-commutative 96.7%

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

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

Alternative 9: 78.6% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. associate-*r* 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 96.6%

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

   1. +-commutative 96.6%

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

   2. fma-define 96.8%

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

   3. +-commutative 96.8%

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

   4. mul-1-neg 96.8%

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

   5. unsub-neg 96.8%

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

   6. associate-*r* 96.8%

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

   7. *-commutative 96.8%

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

8. Simplified 96.8%

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

9. Taylor expanded in eps around 0 76.6%

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

    1. mul-1-neg 76.6%

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

    2. distribute-rgt-neg-in 76.6%

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

11. Simplified 76.6%

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

12. Final simplification 76.6%

    \[\leadsto \varepsilon \cdot \left(-x\right) \]
13. 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 2024131 
(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)))