2sin (example 3.3)

Percentage Accurate: 62.2% → 100.0%

Time: 13.8s

Alternatives: 6

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: 62.2% 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

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = ((cos(x) * cos((eps * 0.5d0))) - (sin(x) * t_0)) * (t_0 * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = ((cos(x) * cos((eps * 0.5))) - (sin(x) * t_0)) * (t_0 * 2.0);
end

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. diff-sin 64.2%

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

   2. div-inv 64.2%

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

   3. +-commutative 64.2%

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

   4. associate--l+ 99.9%

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

   5. metadata-eval 99.9%

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

   6. div-inv 99.9%

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

   7. +-commutative 99.9%

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

   8. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*r* 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-+r+ 99.9%

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

   5. +-commutative 99.9%

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

   6. remove-double-neg 99.9%

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

   7. mul-1-neg 99.9%

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

   8. *-rgt-identity 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. metadata-eval 99.9%

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

   12. distribute-lft-out 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-inverses 99.9%

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

   15. add0 99.9%

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

6. Simplified 99.9%

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

   1. distribute-lft-in 99.9%

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

   2. *-commutative 99.9%

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

   3. cos-sum 100.0%

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

   4. *-commutative 100.0%

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

   5. *-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. metadata-eval 100.0%

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

   6. *-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. *-commutative 100.0%

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

   9. associate-*l* 100.0%

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

   10. metadata-eval 100.0%

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

   11. *-rgt-identity 100.0%

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

   12. *-commutative 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. diff-sin 64.2%

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

   2. div-inv 64.2%

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

   3. +-commutative 64.2%

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

   4. associate--l+ 99.9%

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

   5. metadata-eval 99.9%

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

   6. div-inv 99.9%

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

   7. +-commutative 99.9%

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

   8. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*r* 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-+r+ 99.9%

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

   5. +-commutative 99.9%

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

   6. remove-double-neg 99.9%

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

   7. mul-1-neg 99.9%

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

   8. *-rgt-identity 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-neg-in 99.9%

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

   11. metadata-eval 99.9%

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

   12. distribute-lft-out 99.9%

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

   13. metadata-eval 99.9%

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

   14. +-inverses 99.9%

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

   15. add0 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in eps around inf 99.9%

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

   1. +-commutative 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*l* 99.9%

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

   5. metadata-eval 99.9%

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

   6. *-rgt-identity 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 99.0%

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

   1. unpow2 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

4. Final simplification 98.9%

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

Alternative 5: 97.6% 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 64.1%

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

3. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.1%

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

Alternative 6: 97.6% 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 64.1%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \varepsilon \]
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 2024034 
(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)))