2sin (example 3.3)

Percentage Accurate: 62.4% → 99.8%

Time: 16.5s

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: 62.4% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* (sin eps) (cos x))
  (* (* eps eps) (* (sin x) (fma (* eps eps) 0.041666666666666664 -0.5)))))

C

double code(double x, double eps) {
	return (sin(eps) * cos(x)) + ((eps * eps) * (sin(x) * fma((eps * eps), 0.041666666666666664, -0.5)));
}

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) + Float64(Float64(eps * eps) * Float64(sin(x) * fma(Float64(eps * eps), 0.041666666666666664, -0.5))))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. sin-sum 63.8%

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

   2. associate--l+ 63.8%

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

4. Applied egg-rr 63.8%

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

   1. +-commutative 63.8%

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

   2. sub-neg 63.8%

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

   3. sub0-neg 63.8%

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

   4. associate-+l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. sub0-neg 99.4%

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

   7. neg-mul-1 99.4%

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

   8. *-commutative 99.4%

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

   9. distribute-lft-out 99.5%

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

   10. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in eps around 0 100.0%

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{{\varepsilon}^{2} \cdot \left(-0.5 \cdot \sin x + 0.041666666666666664 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(-0.5 \cdot \sin x + 0.041666666666666664 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) \]

   2. associate-*r* 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.5 \cdot \sin x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right) \]

   3. distribute-rgt-out 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\sin x \cdot \left(-0.5 + 0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \]

   4. +-commutative 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{2} + -0.5\right)}\right) \]

   5. *-commutative 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot 0.041666666666666664} + -0.5\right)\right) \]

   6. fma-define 100.0%

      \[\leadsto \sin \varepsilon \cdot \cos x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, 0.041666666666666664, -0.5\right)}\right) \]

   7. unpow2 100.0%

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

9. Simplified 100.0%

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.041666666666666664, -0.5\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. diff-sin 63.7%

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

   2. *-commutative 63.7%

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

   3. div-inv 63.7%

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

   4. associate--l+ 63.7%

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

   5. metadata-eval 63.7%

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

   6. div-inv 63.7%

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

   7. +-commutative 63.7%

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

   8. associate-+l+ 63.7%

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

   9. metadata-eval 63.7%

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

4. Applied egg-rr 63.7%

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

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%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. distribute-rgt-in 99.4%

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

   3. *-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 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 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

Alternative 5: 99.1% 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 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 98.8%

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

Alternative 6: 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 63.7%

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

3. Taylor expanded in eps around 0 98.8%

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

Alternative 7: 98.5% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 97.9%

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

   1. distribute-rgt-in 97.9%

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

   2. *-un-lft-identity 97.9%

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

   3. distribute-lft-out 97.9%

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

6. Applied egg-rr 97.9%

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

   1. associate-*l* 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

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

Alternative 8: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 97.9%

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

5. Taylor expanded in x around inf 97.9%

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

   1. unpow2 97.9%

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

7. Simplified 97.9%

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

Alternative 9: 98.0% 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 63.7%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024097 
(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)))