2sin (example 3.3)

Percentage Accurate: 62.3% → 99.9%

Time: 13.6s

Alternatives: 8

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: 62.3% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Sin[N[(0.5 * eps), $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. *-commutative 62.8%

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

   3. div-inv 62.8%

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

   4. associate--l+ 62.8%

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

   5. metadata-eval 62.8%

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

   6. div-inv 62.8%

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

   7. +-commutative 62.8%

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

   8. associate-+l+ 62.8%

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

   9. metadata-eval 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in x around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. *-lft-identity 99.8%

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

   4. metadata-eval 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. sub-neg 99.8%

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

   7. mul-1-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) + (eps * (-0.16666666666666666d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(eps * N[(N[(-0.5 * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(eps * -0.16666666666666666), $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 99.6%

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

4. Taylor expanded in x around 0 99.4%

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

   1. *-commutative 99.4%

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

6. Simplified 99.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(-0.5 * N[(eps * N[Sin[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 99.3%

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

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 * eps), $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. *-commutative 62.8%

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

   3. div-inv 62.8%

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

   4. associate--l+ 62.8%

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

   5. metadata-eval 62.8%

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

   6. div-inv 62.8%

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

   7. +-commutative 62.8%

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

   8. associate-+l+ 62.8%

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

   9. metadata-eval 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in x around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. *-lft-identity 99.8%

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

   4. metadata-eval 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. sub-neg 99.8%

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

   7. mul-1-neg 99.8%

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

   8. remove-double-neg 99.8%

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

   9. +-commutative 99.8%

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

   10. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0 99.3%

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

   1. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

    \[\leadsto 2 \cdot \left(\cos \left(0.5 \cdot \left(\varepsilon + x \cdot 2\right)\right) \cdot \left(0.5 \cdot \varepsilon\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(-0.5 * N[(eps * x), $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 99.3%

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

4. Taylor expanded in x around 0 98.4%

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

Alternative 6: 99.0% 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 98.3%

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

Alternative 7: 98.3% 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 99.3%

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

4. Taylor expanded in x around 0 97.0%

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

   1. distribute-lft-out 97.0%

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

6. Simplified 97.0%

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

Alternative 8: 97.9% 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 x around 0 96.5%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

4. Taylor expanded in eps around 0 96.5%

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

Developer Target 1: 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 2024119 
(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
  (! :herbie-platform default (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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