2sin (example 3.3)

Percentage Accurate: 62.7% → 99.7%

Time: 13.7s

Alternatives: 12

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: 62.7% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.125 + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x \cdot 0.5\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (cos x)
   (*
    eps
    (-
     (* eps (+ (* (cos x) -0.125) (* 0.020833333333333332 (* eps (sin x)))))
     (* (sin x) 0.5))))
  (* 2.0 (sin (* eps 0.5)))))

C

double code(double x, double eps) {
	return (cos(x) + (eps * ((eps * ((cos(x) * -0.125) + (0.020833333333333332 * (eps * sin(x))))) - (sin(x) * 0.5)))) * (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 * ((eps * ((cos(x) * (-0.125d0)) + (0.020833333333333332d0 * (eps * sin(x))))) - (sin(x) * 0.5d0)))) * (2.0d0 * sin((eps * 0.5d0)))
end function

Java

public static double code(double x, double eps) {
	return (Math.cos(x) + (eps * ((eps * ((Math.cos(x) * -0.125) + (0.020833333333333332 * (eps * Math.sin(x))))) - (Math.sin(x) * 0.5)))) * (2.0 * Math.sin((eps * 0.5)));
}

Python

def code(x, eps):
	return (math.cos(x) + (eps * ((eps * ((math.cos(x) * -0.125) + (0.020833333333333332 * (eps * math.sin(x))))) - (math.sin(x) * 0.5)))) * (2.0 * math.sin((eps * 0.5)))

Julia

function code(x, eps)
	return Float64(Float64(cos(x) + Float64(eps * Float64(Float64(eps * Float64(Float64(cos(x) * -0.125) + Float64(0.020833333333333332 * Float64(eps * sin(x))))) - Float64(sin(x) * 0.5)))) * Float64(2.0 * sin(Float64(eps * 0.5))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) + (eps * ((eps * ((cos(x) * -0.125) + (0.020833333333333332 * (eps * sin(x))))) - (sin(x) * 0.5)))) * (2.0 * sin((eps * 0.5)));
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] + N[(eps * N[(N[(eps * N[(N[(N[Cos[x], $MachinePrecision] * -0.125), $MachinePrecision] + N[(0.020833333333333332 * N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

   1. diff-sin 61.1%

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

   2. div-inv 61.1%

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

   3. associate--l+ 61.1%

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

   4. metadata-eval 61.1%

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

   5. div-inv 61.1%

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

   6. +-commutative 61.1%

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

   7. associate-+l+ 61.1%

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

   8. metadata-eval 61.1%

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

4. Applied egg-rr 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. *-commutative 61.1%

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

   4. +-commutative 61.1%

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

   5. count-2 61.1%

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

   6. fma-define 61.1%

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

   7. associate-+r- 61.1%

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

   8. +-commutative 61.1%

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

   9. associate--l+ 99.9%

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

   10. +-inverses 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in eps around 0 99.9%

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

8. Taylor expanded in eps around 0 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.125 + 0.020833333333333332 \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x \cdot 0.5\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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)) + ((eps * sin(x)) * 0.041666666666666664d0))))))
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) + ((eps * Math.sin(x)) * 0.041666666666666664))))));
}

Python

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

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(Float64(eps * sin(x)) * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (cos(x) + (eps * ((sin(x) * -0.5) + (eps * ((cos(x) * -0.16666666666666666) + ((eps * sin(x)) * 0.041666666666666664))))));
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[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. diff-sin 61.1%

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

   2. div-inv 61.1%

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

   3. associate--l+ 61.1%

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

   4. metadata-eval 61.1%

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

   5. div-inv 61.1%

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

   6. +-commutative 61.1%

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

   7. associate-+l+ 61.1%

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

   8. metadata-eval 61.1%

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

4. Applied egg-rr 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. *-commutative 61.1%

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

   4. +-commutative 61.1%

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

   5. count-2 61.1%

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

   6. fma-define 61.1%

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

   7. associate-+r- 61.1%

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

   8. +-commutative 61.1%

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

   9. associate--l+ 99.9%

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

   10. +-inverses 99.9%

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

6. Simplified 99.9%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right)} \]
7. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right)\right)}^{1}} \]

   2. +-rgt-identity 99.9%

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

   3. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\right)}^{1}} \]
9. Step-by-step derivation

   1. unpow1 99.9%

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

   2. *-commutative 99.9%

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

   3. fma-undefine 99.9%

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

   4. distribute-lft-out 99.9%

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

   5. associate-*r* 99.9%

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

   6. metadata-eval 99.9%

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

   7. *-lft-identity 99.9%

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

   8. +-commutative 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. diff-sin 61.1%

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

   2. div-inv 61.1%

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

   3. associate--l+ 61.1%

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

   4. metadata-eval 61.1%

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

   5. div-inv 61.1%

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

   6. +-commutative 61.1%

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

   7. associate-+l+ 61.1%

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

   8. metadata-eval 61.1%

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

4. Applied egg-rr 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. *-commutative 61.1%

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

   4. +-commutative 61.1%

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

   5. count-2 61.1%

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

   6. fma-define 61.1%

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

   7. associate-+r- 61.1%

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

   8. +-commutative 61.1%

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

   9. associate--l+ 99.9%

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

   10. +-inverses 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in eps around 0 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.9%

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

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. Taylor expanded in x around 0 99.0%

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

6. Final simplification 99.0%

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

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 98.9%

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

   1. associate-*r* 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 7: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.9%

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

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. Taylor expanded in x around 0 98.8%

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

   1. *-commutative 98.8%

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

7. Simplified 98.8%

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

Alternative 8: 99.0% 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 61.1%

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

3. Taylor expanded in eps around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 9: 98.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.08333333333333333 \cdot \left(x \cdot \varepsilon\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (* x (+ (* eps -0.5) (* x (- (* 0.08333333333333333 (* x eps)) 0.5)))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (x * eps)) - 0.5)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * ((eps * (-0.5d0)) + (x * ((0.08333333333333333d0 * (x * eps)) - 0.5d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (x * eps)) - 0.5)))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (x * eps)) - 0.5)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.08333333333333333 * Float64(x * eps)) - 0.5))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * ((eps * -0.5) + (x * ((0.08333333333333333 * (x * eps)) - 0.5)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.08333333333333333 * N[(x * eps), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.1%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(-0.5 \cdot \varepsilon + x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon \cdot x\right) - 0.5\right)\right)\right)} \]

5. Final simplification 97.1%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.08333333333333333 \cdot \left(x \cdot \varepsilon\right) - 0.5\right)\right)\right) \]
6. Add Preprocessing

Alternative 10: 98.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.1%

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

   1. distribute-lft-out 97.1%

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

   2. +-commutative 97.1%

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

6. Simplified 97.1%

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

Alternative 11: 98.2% 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 61.1%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 97.1%

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

   1. distribute-lft-out 97.1%

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

   2. +-commutative 97.1%

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

6. Simplified 97.1%

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

7. Taylor expanded in x around inf 97.0%

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

   1. *-commutative 97.0%

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

9. Simplified 97.0%

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

Alternative 12: 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 61.1%

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

3. Taylor expanded in x around 0 96.4%

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

4. Taylor expanded in eps around 0 96.4%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024165 
(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)))