2sin (example 3.3)

Percentage Accurate: 62.3% → 100.0%

Time: 14.9s

Alternatives: 13

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: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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 = 2.0d0 * (t_0 * ((cos((eps * 0.5d0)) * cos(x)) - (t_0 * sin(x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. diff-sin 62.9%

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

   2. div-inv 62.9%

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

   3. associate--l+ 62.9%

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

   4. metadata-eval 62.9%

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

   5. div-inv 62.9%

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

   6. +-commutative 62.9%

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

   7. associate-+l+ 62.9%

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

   8. metadata-eval 62.9%

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

4. Applied egg-rr 62.9%

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

   1. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

   3. *-commutative 62.9%

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

   4. +-commutative 62.9%

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

   5. count-2 62.9%

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

   6. fma-define 62.9%

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

   7. associate-+r- 62.9%

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

   8. +-commutative 62.9%

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

   9. associate--l+ 99.5%

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

   10. +-inverses 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around inf 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. cancel-sign-sub-inv 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. *-lft-identity 99.5%

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

   8. metadata-eval 99.5%

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

   9. cancel-sign-sub-inv 99.5%

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

   10. *-commutative 99.5%

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

   11. sub-neg 99.5%

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

   12. mul-1-neg 99.5%

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

   13. remove-double-neg 99.5%

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

9. Simplified 99.5%

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

    1. +-commutative 99.5%

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

    2. cos-sum 100.0%

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

11. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5 + \varepsilon \cdot \left(\cos x \cdot -0.16666666666666666 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (cos x)
   (*
    eps
    (+
     (* (sin x) -0.5)
     (*
      eps
      (+
       (* (cos x) -0.16666666666666666)
       (* 0.041666666666666664 (* eps (sin x))))))))))

C

double code(double x, double eps) {
	return eps * (cos(x) + (eps * ((sin(x) * -0.5) + (eps * ((cos(x) * -0.16666666666666666) + (0.041666666666666664 * (eps * sin(x))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (cos(x) + (eps * ((sin(x) * (-0.5d0)) + (eps * ((cos(x) * (-0.16666666666666666d0)) + (0.041666666666666664d0 * (eps * sin(x))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.cos(x) + (eps * ((Math.sin(x) * -0.5) + (eps * ((Math.cos(x) * -0.16666666666666666) + (0.041666666666666664 * (eps * Math.sin(x))))))));
}

Python

def code(x, eps):
	return eps * (math.cos(x) + (eps * ((math.sin(x) * -0.5) + (eps * ((math.cos(x) * -0.16666666666666666) + (0.041666666666666664 * (eps * math.sin(x))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(cos(x) + Float64(eps * Float64(Float64(sin(x) * -0.5) + Float64(eps * Float64(Float64(cos(x) * -0.16666666666666666) + Float64(0.041666666666666664 * Float64(eps * sin(x)))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (cos(x) + (eps * ((sin(x) * -0.5) + (eps * ((cos(x) * -0.16666666666666666) + (0.041666666666666664 * (eps * sin(x))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * -0.5), $MachinePrecision] + N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(0.041666666666666664 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(-0.5 \cdot \sin x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

4. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5 + \varepsilon \cdot \left(\cos x \cdot -0.16666666666666666 + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\cos x + \varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.125\right) - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+ (cos x) (* eps (- (* (cos x) (* eps -0.125)) (* 0.5 (sin x)))))
  (* 2.0 (sin (* eps 0.5)))))

C

double code(double x, double eps) {
	return (cos(x) + (eps * ((cos(x) * (eps * -0.125)) - (0.5 * sin(x))))) * (2.0 * sin((eps * 0.5)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos(x) + (eps * ((cos(x) * (eps * (-0.125d0))) - (0.5d0 * sin(x))))) * (2.0d0 * sin((eps * 0.5d0)))
end function

Java

public static double code(double x, double eps) {
	return (Math.cos(x) + (eps * ((Math.cos(x) * (eps * -0.125)) - (0.5 * Math.sin(x))))) * (2.0 * Math.sin((eps * 0.5)));
}

Python

def code(x, eps):
	return (math.cos(x) + (eps * ((math.cos(x) * (eps * -0.125)) - (0.5 * math.sin(x))))) * (2.0 * math.sin((eps * 0.5)))

Julia

function code(x, eps)
	return Float64(Float64(cos(x) + Float64(eps * Float64(Float64(cos(x) * Float64(eps * -0.125)) - Float64(0.5 * sin(x))))) * Float64(2.0 * sin(Float64(eps * 0.5))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) + (eps * ((cos(x) * (eps * -0.125)) - (0.5 * sin(x))))) * (2.0 * sin((eps * 0.5)));
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] + N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * -0.125), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. diff-sin 62.9%

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

   2. div-inv 62.9%

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

   3. associate--l+ 62.9%

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

   4. metadata-eval 62.9%

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

   5. div-inv 62.9%

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

   6. +-commutative 62.9%

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

   7. associate-+l+ 62.9%

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

   8. metadata-eval 62.9%

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

4. Applied egg-rr 62.9%

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

   1. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

   3. *-commutative 62.9%

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

   4. +-commutative 62.9%

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

   5. count-2 62.9%

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

   6. fma-define 62.9%

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

   7. associate-+r- 62.9%

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

   8. +-commutative 62.9%

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

   9. associate--l+ 99.5%

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

   10. +-inverses 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{\left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.125 \cdot \cos x + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - 0.5 \cdot \sin x\right)\right)} \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right) \]

8. Taylor expanded in eps around 0 100.0%

   \[\leadsto \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.125 \cdot \cos x + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right) \]
9. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.125 \cdot \cos x + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \]

10. Simplified 100.0%

    \[\leadsto \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.125 \cdot \cos x + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \]

11. Taylor expanded in eps around 0 99.9%

    \[\leadsto \left(\cos x + \varepsilon \cdot \left(\color{blue}{-0.125 \cdot \left(\varepsilon \cdot \cos x\right)} - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
12. Step-by-step derivation

    1. associate-*r* 99.9%

       \[\leadsto \left(\cos x + \varepsilon \cdot \left(\color{blue}{\left(-0.125 \cdot \varepsilon\right) \cdot \cos x} - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

    2. *-commutative 99.9%

       \[\leadsto \left(\cos x + \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot -0.125\right)} \cdot \cos x - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

13. Simplified 99.9%

    \[\leadsto \left(\cos x + \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot -0.125\right) \cdot \cos x} - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]

14. Final simplification 99.9%

    \[\leadsto \left(\cos x + \varepsilon \cdot \left(\cos x \cdot \left(\varepsilon \cdot -0.125\right) - 0.5 \cdot \sin x\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
15. Add Preprocessing

Alternative 4: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (cos(x) + (eps * ((sin(x) * (-0.5d0)) + ((-0.16666666666666666d0) * (eps * cos(x))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (cos(x) + (eps * ((sin(x) * -0.5) + (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 * ((sin(x) * (-0.5d0)) + (eps * (-0.16666666666666666d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(-0.5 \cdot \sin x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

4. Taylor expanded in x around 0 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\sin x \cdot -0.5 + \varepsilon \cdot -0.16666666666666666\right)\right) \]
6. Add Preprocessing

Alternative 6: 99.4% 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 62.9%

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

3. Taylor expanded in eps around 0 99.7%

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

Alternative 7: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. diff-sin 62.9%

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

   2. div-inv 62.9%

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

   3. associate--l+ 62.9%

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

   4. metadata-eval 62.9%

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

   5. div-inv 62.9%

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

   6. +-commutative 62.9%

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

   7. associate-+l+ 62.9%

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

   8. metadata-eval 62.9%

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

4. Applied egg-rr 62.9%

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

   1. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

   3. *-commutative 62.9%

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

   4. +-commutative 62.9%

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

   5. count-2 62.9%

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

   6. fma-define 62.9%

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

   7. associate-+r- 62.9%

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

   8. +-commutative 62.9%

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

   9. associate--l+ 99.5%

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

   10. +-inverses 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around inf 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. cancel-sign-sub-inv 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. *-lft-identity 99.5%

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

   8. metadata-eval 99.5%

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

   9. cancel-sign-sub-inv 99.5%

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

   10. *-commutative 99.5%

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

   11. sub-neg 99.5%

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

   12. mul-1-neg 99.5%

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

   13. remove-double-neg 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around 0 99.4%

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

11. Final simplification 99.4%

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

Alternative 8: 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 62.9%

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

3. Taylor expanded in eps around 0 98.5%

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

Alternative 9: 98.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.7%

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

4. Taylor expanded in x around 0 96.4%

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

   1. distribute-lft-out 96.4%

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

   2. unpow2 96.4%

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

   3. distribute-lft-out 96.4%

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

6. Simplified 96.4%

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

7. Final simplification 96.4%

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

Alternative 10: 98.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.7%

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

4. Taylor expanded in x around 0 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in x around 0 96.4%

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

   1. distribute-rgt-in 96.4%

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

   2. associate-*r* 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-*r* 96.4%

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

   5. *-commutative 96.4%

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

   6. *-commutative 96.4%

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

   7. distribute-lft-out 96.4%

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

   8. *-commutative 96.4%

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

9. Simplified 96.4%

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

10. Final simplification 96.4%

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

Alternative 11: 98.1% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.7%

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

4. Taylor expanded in x around 0 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in x around 0 96.4%

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

   1. distribute-rgt-in 96.4%

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

   2. associate-*r* 96.4%

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

   3. *-commutative 96.4%

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

   4. associate-*r* 96.4%

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

   5. *-commutative 96.4%

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

   6. *-commutative 96.4%

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

   7. distribute-lft-out 96.4%

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

   8. *-commutative 96.4%

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

9. Simplified 96.4%

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

10. Taylor expanded in eps around 0 96.3%

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

11. Final simplification 96.3%

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

Alternative 12: 97.7% accurate, 22.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 + (-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 + ((-0.5d0) * (eps * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in eps around 0 99.7%

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

4. Taylor expanded in x around 0 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in x around 0 95.8%

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

8. Final simplification 95.8%

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

Alternative 13: 97.7% 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 62.9%

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

3. Taylor expanded in x around 0 95.8%

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

4. Taylor expanded in eps around 0 95.8%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024112 
(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)))