2cos (problem 3.3.5)

Percentage Accurate: 52.3% → 99.5%

Time: 21.8s

Alternatives: 10

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: 99.5% 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 54.1%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (sin (* eps 0.5))
  (* -2.0 (expm1 (log1p (sin (* 0.5 (- eps (* x -2.0)))))))))

C

double code(double x, double eps) {
	return sin((eps * 0.5)) * (-2.0 * expm1(log1p(sin((0.5 * (eps - (x * -2.0)))))));
}

Java

public static double code(double x, double eps) {
	return Math.sin((eps * 0.5)) * (-2.0 * Math.expm1(Math.log1p(Math.sin((0.5 * (eps - (x * -2.0)))))));
}

Python

def code(x, eps):
	return math.sin((eps * 0.5)) * (-2.0 * math.expm1(math.log1p(math.sin((0.5 * (eps - (x * -2.0)))))))

Julia

function code(x, eps)
	return Float64(sin(Float64(eps * 0.5)) * Float64(-2.0 * expm1(log1p(sin(Float64(0.5 * Float64(eps - Float64(x * -2.0))))))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[(Exp[N[Log[1 + N[Sin[N[(0.5 * N[(eps - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

   1. diff-cos 80.8%

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

   2. div-inv 80.8%

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

   3. associate--l+ 80.8%

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

   4. metadata-eval 80.8%

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

   5. div-inv 80.8%

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

   6. +-commutative 80.8%

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

   7. associate-+l+ 80.8%

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

   8. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

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

   1. associate-*r* 80.8%

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

   2. *-commutative 80.8%

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

   3. associate-*l* 80.8%

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

   4. associate-+r- 80.8%

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

   5. +-commutative 80.8%

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

   6. associate--l+ 99.5%

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

   7. +-inverses 99.5%

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

   8. +-commutative 99.5%

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

   9. *-lft-identity 99.5%

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

   10. metadata-eval 99.5%

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

   11. cancel-sign-sub-inv 99.5%

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

   12. neg-sub0 99.5%

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

   13. mul-1-neg 99.5%

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

   14. remove-double-neg 99.5%

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

   15. *-commutative 99.5%

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

   16. +-commutative 99.5%

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

   17. count-2 99.5%

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

   18. fma-define 99.5%

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

6. Simplified 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\right)}\right) \]

   2. expm1-undefine 57.7%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)} - 1\right)}\right) \]

8. Applied egg-rr 57.7%

   \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)} - 1\right)}\right) \]
9. Step-by-step derivation

   1. expm1-define 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right)\right)}\right) \]

   2. fma-undefine 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right)\right)\right)\right) \]

   3. +-commutative 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + 2 \cdot x\right)}\right)\right)\right)\right) \]

   4. metadata-eval 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{\left(--2\right)} \cdot x\right)\right)\right)\right)\right) \]

   5. cancel-sign-sub-inv 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon - -2 \cdot x\right)}\right)\right)\right)\right) \]

   6. cancel-sign-sub-inv 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(--2\right) \cdot x\right)}\right)\right)\right)\right) \]

   7. metadata-eval 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\varepsilon + \color{blue}{2} \cdot x\right)\right)\right)\right)\right) \]

   8. +-commutative 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right)\right)\right)\right) \]

   9. *-commutative 99.5%

      \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\color{blue}{x \cdot 2} + \varepsilon\right)\right)\right)\right)\right) \]

   10. fma-undefine 99.5%

       \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, 2, \varepsilon\right)}\right)\right)\right)\right) \]

10. Simplified 99.5%

    \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \mathsf{fma}\left(x, 2, \varepsilon\right)\right)\right)\right)}\right) \]

11. Taylor expanded in x around -inf 99.5%

    \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)}\right)\right)\right) \]

12. Final simplification 99.5%

    \[\leadsto \sin \left(\varepsilon \cdot 0.5\right) \cdot \left(-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return sin((eps * 0.5)) * (-2.0 * sin((0.5 * (eps - (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)) * ((-2.0d0) * sin((0.5d0 * (eps - (x * (-2.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. diff-cos 80.8%

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

   2. div-inv 80.8%

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

   3. associate--l+ 80.8%

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

   4. metadata-eval 80.8%

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

   5. div-inv 80.8%

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

   6. +-commutative 80.8%

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

   7. associate-+l+ 80.8%

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

   8. metadata-eval 80.8%

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

4. Applied egg-rr 80.8%

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

   1. associate-*r* 80.8%

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

   2. *-commutative 80.8%

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

   3. associate-*l* 80.8%

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

   4. associate-+r- 80.8%

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

   5. +-commutative 80.8%

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

   6. associate--l+ 99.5%

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

   7. +-inverses 99.5%

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

   8. +-commutative 99.5%

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

   9. *-lft-identity 99.5%

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

   10. metadata-eval 99.5%

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

   11. cancel-sign-sub-inv 99.5%

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

   12. neg-sub0 99.5%

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

   13. mul-1-neg 99.5%

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

   14. remove-double-neg 99.5%

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

   15. *-commutative 99.5%

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

   16. +-commutative 99.5%

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

   17. count-2 99.5%

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

   18. fma-define 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around -inf 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 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 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2.25e-14)
   (*
    eps
    (+
     (* eps -0.5)
     (* x (+ (* x (+ (* x 0.16666666666666666) (* eps 0.25))) -1.0))))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2.25e-14) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2.25d-14) then
        tmp = eps * ((eps * (-0.5d0)) + (x * ((x * ((x * 0.16666666666666666d0) + (eps * 0.25d0))) + (-1.0d0))))
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2.25e-14) {
		tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2.25e-14:
		tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)))
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2.25e-14)
		tmp = Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(eps * 0.25))) + -1.0))));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2.25e-14)
		tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2.25e-14], 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], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.2499999999999999e-14

   1. Initial program 54.1%

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

   3. Taylor expanded in eps around 0 99.8%

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

      1. associate-*r* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

   if 2.2499999999999999e-14 < eps

   1. Initial program 53.6%

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

   3. Taylor expanded in eps around 0 53.2%

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

      1. mul-1-neg 53.2%

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

      2. *-commutative 53.2%

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

      3. distribute-rgt-neg-in 53.2%

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

   5. Simplified 53.2%

      \[\leadsto \color{blue}{\sin x \cdot \left(-\varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

   1. add-cube-cbrt 98.7%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(\sqrt[3]{\left(-0.5 \cdot \varepsilon\right) \cdot \cos x} \cdot \sqrt[3]{\left(-0.5 \cdot \varepsilon\right) \cdot \cos x}\right) \cdot \sqrt[3]{\left(-0.5 \cdot \varepsilon\right) \cdot \cos x}} - \sin x\right) \]

   2. pow3 98.7%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{{\left(\sqrt[3]{\left(-0.5 \cdot \varepsilon\right) \cdot \cos x}\right)}^{3}} - \sin x\right) \]

   3. *-commutative 98.7%

      \[\leadsto \varepsilon \cdot \left({\left(\sqrt[3]{\color{blue}{\cos x \cdot \left(-0.5 \cdot \varepsilon\right)}}\right)}^{3} - \sin x\right) \]

   4. *-commutative 98.7%

      \[\leadsto \varepsilon \cdot \left({\left(\sqrt[3]{\cos x \cdot \color{blue}{\left(\varepsilon \cdot -0.5\right)}}\right)}^{3} - \sin x\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos x \cdot \left(\varepsilon \cdot -0.5\right)}\right)}^{3}} - \sin x\right) \]

8. Taylor expanded in x around 0 97.6%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot {\left(\sqrt[3]{-0.5}\right)}^{3}} - \sin x\right) \]
9. Step-by-step derivation

   1. rem-cube-cbrt 97.6%

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

10. Simplified 97.6%

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

11. Final simplification 97.6%

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

Alternative 7: 98.1% 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 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.2%

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

7. Final simplification 95.2%

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

Alternative 8: 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 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.1%

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

7. Final simplification 95.1%

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

Alternative 9: 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 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.1%

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

   1. +-commutative 95.1%

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

   2. *-commutative 95.1%

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

   3. mul-1-neg 95.1%

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

   4. unsub-neg 95.1%

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

8. Simplified 95.1%

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

9. Final simplification 95.1%

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

Alternative 10: 78.5% 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 54.1%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. associate-*r* 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 95.1%

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

   1. associate-*r* 95.1%

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

   2. +-commutative 95.1%

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

   3. associate-*r* 95.1%

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

   4. mul-1-neg 95.1%

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

   5. unsub-neg 95.1%

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

   6. *-commutative 95.1%

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

   7. *-commutative 95.1%

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

8. Simplified 95.1%

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

9. Taylor expanded in eps around 0 76.9%

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

    1. mul-1-neg 76.9%

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

    2. distribute-rgt-neg-out 76.9%

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

11. Simplified 76.9%

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

12. Final simplification 76.9%

    \[\leadsto \varepsilon \cdot \left(-x\right) \]
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 2024053 
(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)))