2sin (example 3.3)

Percentage Accurate: 62.1% → 100.0%

Time: 15.9s

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: 62.1% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

4. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. metadata-eval 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon} \]

   3. 1-sub-cos 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   4. pow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

8. Applied egg-rr 100.0%

   \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   2. associate-/l* 100.0%

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

   3. hang-0p-tan 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

4. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. metadata-eval 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon} \]

   3. 1-sub-cos 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   4. pow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

8. Applied egg-rr 100.0%

   \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   2. associate-/l* 100.0%

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

   3. hang-0p-tan 100.0%

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

10. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. associate-*r* 100.0%

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

    3. div-inv 100.0%

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

    4. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

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

    1. sub-neg 100.0%

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

    2. *-commutative 100.0%

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

    3. *-commutative 100.0%

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

    4. associate-*r* 100.0%

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

    5. distribute-rgt-out-- 100.0%

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

    6. *-commutative 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + x)))))
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 + x)))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

   1. diff-sin 63.3%

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

   2. *-commutative 63.3%

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

   3. div-inv 63.3%

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

   4. associate--l+ 63.3%

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

   5. metadata-eval 63.3%

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

   6. div-inv 63.3%

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

   7. +-commutative 63.3%

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

   8. associate-+l+ 63.3%

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

   9. metadata-eval 63.3%

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

4. Applied egg-rr 63.3%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in eps around 0 98.8%

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

4. Taylor expanded in x around 0 97.9%

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

   1. associate-*r* 97.9%

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

   2. *-commutative 97.9%

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

   3. *-commutative 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

   \[\leadsto x \cdot \left({\varepsilon}^{2} \cdot -0.5\right) + \varepsilon \cdot \cos x \]
8. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

4. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. metadata-eval 99.3%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon} \]

   3. 1-sub-cos 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   4. pow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

8. Applied egg-rr 100.0%

   \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon} \]

   2. associate-/l* 100.0%

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

   3. hang-0p-tan 100.0%

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

10. Simplified 100.0%

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

    1. sub-neg 100.0%

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

    2. associate-*r* 100.0%

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

    3. div-inv 100.0%

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

    4. metadata-eval 100.0%

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

12. Applied egg-rr 100.0%

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

    1. sub-neg 100.0%

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

    2. *-commutative 100.0%

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

    3. *-commutative 100.0%

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

    4. associate-*r* 100.0%

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

    5. distribute-rgt-out-- 100.0%

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

    6. *-commutative 100.0%

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

14. Simplified 100.0%

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

15. Taylor expanded in eps around 0 97.9%

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

16. Final simplification 97.9%

    \[\leadsto \sin \varepsilon \cdot \cos x \]
17. 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 63.3%

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

3. Taylor expanded in eps around 0 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \varepsilon \cdot \cos x \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in x around 0 95.8%

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

4. Final simplification 95.8%

   \[\leadsto \sin \varepsilon \]
5. Add Preprocessing

Alternative 8: 97.8% 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 63.3%

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

3. Taylor expanded in eps around 0 97.9%

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

4. Taylor expanded in x around 0 95.8%

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

5. Final simplification 95.8%

   \[\leadsto \varepsilon \]
6. 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 2024040 
(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)))

  :herbie-target
  (* (* 2.0 (cos (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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