2sin (example 3.3)

Percentage Accurate: 62.3% → 99.7%

Time: 16.5s

Alternatives: 8

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\sin x \cdot 0.041666666666666664\right) + \cos x \cdot -0.16666666666666666, \sin x \cdot -0.5\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (cos x)
   (log
    (+
     1.0
     (expm1
      (*
       eps
       (fma
        eps
        (+
         (* eps (* (sin x) 0.041666666666666664))
         (* (cos x) -0.16666666666666666))
        (* (sin x) -0.5)))))))))

C

double code(double x, double eps) {
	return eps * (cos(x) + log((1.0 + expm1((eps * fma(eps, ((eps * (sin(x) * 0.041666666666666664)) + (cos(x) * -0.16666666666666666)), (sin(x) * -0.5)))))));
}

Julia

function code(x, eps)
	return Float64(eps * Float64(cos(x) + log(Float64(1.0 + expm1(Float64(eps * fma(eps, Float64(Float64(eps * Float64(sin(x) * 0.041666666666666664)) + Float64(cos(x) * -0.16666666666666666)), Float64(sin(x) * -0.5))))))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[Log[N[(1.0 + N[(Exp[N[(eps * N[(eps * N[(N[(eps * N[(N[Sin[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\varepsilon \cdot \left(-0.5 \cdot \sin x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)\right)}\right) \]

   2. log1p-undefine 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \left(-0.5 \cdot \sin x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)\right)}\right) \]

   3. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right) + -0.5 \cdot \sin x\right)}\right)\right)\right) \]

   4. fma-define 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right), -0.5 \cdot \sin x\right)}\right)\right)\right) \]

   5. fma-define 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \cos x, 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right)}, -0.5 \cdot \sin x\right)\right)\right)\right) \]

   6. *-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-0.16666666666666666, \cos x, 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right), \color{blue}{\sin x \cdot -0.5}\right)\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-0.16666666666666666, \cos x, 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)\right), \sin x \cdot -0.5\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{-0.16666666666666666 \cdot \cos x + 0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right)}, \sin x \cdot -0.5\right)\right)\right)\right) \]

   2. +-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{0.041666666666666664 \cdot \left(\varepsilon \cdot \sin x\right) + -0.16666666666666666 \cdot \cos x}, \sin x \cdot -0.5\right)\right)\right)\right) \]

   3. *-commutative 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \sin x\right) \cdot 0.041666666666666664} + -0.16666666666666666 \cdot \cos x, \sin x \cdot -0.5\right)\right)\right)\right) \]

   4. associate-*l* 100.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\sin x \cdot 0.041666666666666664\right)} + -0.16666666666666666 \cdot \cos x, \sin x \cdot -0.5\right)\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\sin x \cdot 0.041666666666666664\right) + -0.16666666666666666 \cdot \cos x}, \sin x \cdot -0.5\right)\right)\right)\right) \]

8. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\sin x \cdot 0.041666666666666664\right) + \cos x \cdot -0.16666666666666666, \sin x \cdot -0.5\right)\right)\right)\right) \]
9. 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 61.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

   1. diff-sin 61.3%

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

   2. *-commutative 61.3%

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

   3. div-inv 61.3%

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

   4. associate--l+ 61.3%

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

   5. metadata-eval 61.3%

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

   6. div-inv 61.3%

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

   7. +-commutative 61.3%

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

   8. associate-+l+ 61.3%

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

   9. metadata-eval 61.3%

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in x around -inf 99.9%

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

6. Taylor expanded in eps around 0 100.0%

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

7. Final simplification 100.0%

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

Alternative 4: 99.7% 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 61.3%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

   1. diff-sin 61.3%

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

   2. *-commutative 61.3%

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

   3. div-inv 61.3%

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

   4. associate--l+ 61.3%

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

   5. metadata-eval 61.3%

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

   6. div-inv 61.3%

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

   7. +-commutative 61.3%

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

   8. associate-+l+ 61.3%

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

   9. metadata-eval 61.3%

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in x around -inf 99.9%

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

6. Final simplification 99.9%

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

Alternative 6: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Final simplification 99.8%

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

Alternative 7: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 8: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (* (* 2.0 (cos (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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