2sin (example 3.3)

Percentage Accurate: 62.8% → 100.0%

Time: 13.1s

Alternatives: 9

Speedup: 205.0×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ 2 \cdot \left(t\_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \cos x - t\_0 \cdot \sin x\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (*
    2.0
    (*
     t_0
     (log1p (expm1 (- (* (cos (* eps 0.5)) (cos x)) (* t_0 (sin x)))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return 2.0 * (t_0 * log1p(expm1(((cos((eps * 0.5)) * cos(x)) - (t_0 * sin(x))))));
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	return 2.0 * (t_0 * Math.log1p(Math.expm1(((Math.cos((eps * 0.5)) * Math.cos(x)) - (t_0 * Math.sin(x))))));
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return 2.0 * (t_0 * math.log1p(math.expm1(((math.cos((eps * 0.5)) * math.cos(x)) - (t_0 * math.sin(x))))))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(2.0 * Float64(t_0 * log1p(expm1(Float64(Float64(cos(Float64(eps * 0.5)) * cos(x)) - Float64(t_0 * sin(x)))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(2.0 * N[(t$95$0 * N[Log[1 + N[(Exp[N[(N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

   1. diff-sin 62.8%

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

   2. div-inv 62.8%

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

   3. associate--l+ 62.8%

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

   4. metadata-eval 62.8%

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

   5. div-inv 62.8%

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

   6. +-commutative 62.8%

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

   7. associate-+l+ 62.8%

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

   8. metadata-eval 62.8%

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

4. Applied egg-rr 62.8%

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

   1. sub-neg 62.8%

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

   2. mul-1-neg 62.8%

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

   3. +-commutative 62.8%

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

   4. associate-+r+ 99.9%

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

   5. mul-1-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. +-inverses 99.9%

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

   8. remove-double-neg 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. neg-sub0 99.9%

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

   12. mul-1-neg 99.9%

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

   13. remove-double-neg 99.9%

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

   14. *-commutative 99.9%

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

   15. +-commutative 99.9%

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

   16. count-2 99.9%

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

   17. fma-define 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around -inf 99.9%

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

   1. log1p-expm1-u 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

   5. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. distribute-lft-in 99.9%

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

    2. cos-sum 100.0%

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

    3. *-commutative 100.0%

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

    4. associate-*r* 100.0%

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

    5. metadata-eval 100.0%

       \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{1} \cdot x\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)\right)\right) \]

    6. *-un-lft-identity 100.0%

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

    7. *-commutative 100.0%

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

    8. associate-*r* 100.0%

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

    9. metadata-eval 100.0%

       \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\color{blue}{1} \cdot x\right)\right)\right)\right) \]

    10. *-un-lft-identity 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. diff-sin 62.8%

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

   2. div-inv 62.8%

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

   3. associate--l+ 62.8%

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

   4. metadata-eval 62.8%

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

   5. div-inv 62.8%

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

   6. +-commutative 62.8%

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

   7. associate-+l+ 62.8%

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

   8. metadata-eval 62.8%

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

4. Applied egg-rr 62.8%

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

   1. sub-neg 62.8%

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

   2. mul-1-neg 62.8%

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

   3. +-commutative 62.8%

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

   4. associate-+r+ 99.9%

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

   5. mul-1-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. +-inverses 99.9%

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

   8. remove-double-neg 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. neg-sub0 99.9%

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

   12. mul-1-neg 99.9%

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

   13. remove-double-neg 99.9%

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

   14. *-commutative 99.9%

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

   15. +-commutative 99.9%

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

   16. count-2 99.9%

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

   17. fma-define 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around -inf 99.9%

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

   1. log1p-expm1-u 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-rgt-neg-in 99.9%

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

   5. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. diff-sin 62.8%

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

   2. div-inv 62.8%

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

   3. associate--l+ 62.8%

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

   4. metadata-eval 62.8%

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

   5. div-inv 62.8%

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

   6. +-commutative 62.8%

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

   7. associate-+l+ 62.8%

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

   8. metadata-eval 62.8%

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

4. Applied egg-rr 62.8%

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

   1. sub-neg 62.8%

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

   2. mul-1-neg 62.8%

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

   3. +-commutative 62.8%

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

   4. associate-+r+ 99.9%

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

   5. mul-1-neg 99.9%

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

   6. sub-neg 99.9%

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

   7. +-inverses 99.9%

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

   8. remove-double-neg 99.9%

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

   9. mul-1-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. neg-sub0 99.9%

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

   12. mul-1-neg 99.9%

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

   13. remove-double-neg 99.9%

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

   14. *-commutative 99.9%

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

   15. +-commutative 99.9%

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

   16. count-2 99.9%

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

   17. fma-define 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around -inf 99.9%

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

8. Final simplification 99.9%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. associate-*r* 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 98.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Add Preprocessing

Alternative 6: 98.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon \cdot x\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (* x (+ (* eps -0.5) (* x (- (* 0.08333333333333333 (* eps x)) 0.5)))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (eps * x)) - 0.5)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * ((eps * (-0.5d0)) + (x * ((0.08333333333333333d0 * (eps * x)) - 0.5d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (eps * x)) - 0.5)))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (eps * x)) - 0.5)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.08333333333333333 * Float64(eps * x)) - 0.5))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (eps * x)) - 0.5)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.08333333333333333 * N[(eps * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. associate-*r* 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 97.9%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon \cdot x\right) - 0.5\right)\right)\right)} \]

7. Final simplification 97.9%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon \cdot x\right) - 0.5\right)\right)\right) \]
8. Add Preprocessing

Alternative 7: 98.5% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

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 + x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. associate-*r* 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 97.8%

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

   1. distribute-lft-out 97.8%

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

8. Simplified 97.8%

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

Alternative 8: 98.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. associate-*r* 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 97.9%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon \cdot x\right) - 0.5\right)\right)\right)} \]

7. Taylor expanded in eps around 0 97.7%

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

   1. *-commutative 97.7%

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

9. Simplified 97.7%

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

Alternative 9: 98.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. associate-*r* 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 97.8%

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

   1. distribute-lft-out 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in x around 0 97.7%

   \[\leadsto \color{blue}{\varepsilon} \]
10. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (2.0 * cos((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 * cos((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024099 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

  (- (sin (+ x eps)) (sin x)))