2sin (example 3.3)

Percentage Accurate: 62.2% → 99.6%

Time: 14.1s

Alternatives: 15

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \cdot \left(\varepsilon \cdot \cos x\right) + \left(\varepsilon \cdot \sin x\right) \cdot \left(\varepsilon \cdot -0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((1.0d0 + ((eps * eps) * (-0.16666666666666666d0))) * (eps * cos(x))) + ((eps * sin(x)) * (eps * (-0.5d0)))
end function

Java

public static double code(double x, double eps) {
	return ((1.0 + ((eps * eps) * -0.16666666666666666)) * (eps * Math.cos(x))) + ((eps * Math.sin(x)) * (eps * -0.5));
}

Python

def code(x, eps):
	return ((1.0 + ((eps * eps) * -0.16666666666666666)) * (eps * math.cos(x))) + ((eps * math.sin(x)) * (eps * -0.5))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((1.0 + ((eps * eps) * -0.16666666666666666)) * (eps * cos(x))) + ((eps * sin(x)) * (eps * -0.5));
end

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\cos x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) + \cos \color{blue}{x}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right), \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\sin x \cdot \frac{-1}{2}\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \sin x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \cos x\right)}\right) + \cos x\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right) + \cos x\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x + \cos \color{blue}{x}\right)\right)\right) \]

   14. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \color{blue}{\cos x}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right), \color{blue}{\cos x}\right)\right)\right) \]

5. Simplified 100.0%

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

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \cos x + \color{blue}{\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \cos x\right) + \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \cos x\right) + \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \color{blue}{\varepsilon} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \cos x\right)\right), \color{blue}{\left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \cos x\right) \cdot \varepsilon\right), \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \cdot \varepsilon\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \left(\cos x \cdot \varepsilon\right)\right), \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \cdot \varepsilon\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right) \cdot \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right) \cdot \varepsilon\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6} + 1\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \cdot \varepsilon\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6}\right), 1\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\color{blue}{\frac{-1}{2}} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-1}{6}\right), 1\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \mathsf{*.f64}\left(\varepsilon, \cos x\right)\right), \left(\left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right) \cdot \varepsilon\right)\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \color{blue}{\sin x}\right)\right) \cdot \varepsilon\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \frac{-1}{2}\right) \cdot \varepsilon\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right), 1\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \sin x\right), \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}\right)\right) \]

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \cdot \left(\varepsilon \cdot \cos x\right) + \left(\varepsilon \cdot \sin x\right) \cdot \left(\varepsilon \cdot -0.5\right) \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\varepsilon \cdot \sin x\right) \cdot -0.5 + \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \cdot \cos x\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((eps * sin(x)) * (-0.5d0)) + ((1.0d0 + ((eps * eps) * (-0.16666666666666666d0))) * cos(x)))
end function

Java

public static double code(double x, double eps) {
	return eps * (((eps * Math.sin(x)) * -0.5) + ((1.0 + ((eps * eps) * -0.16666666666666666)) * Math.cos(x)));
}

Python

def code(x, eps):
	return eps * (((eps * math.sin(x)) * -0.5) + ((1.0 + ((eps * eps) * -0.16666666666666666)) * math.cos(x)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((eps * sin(x)) * -0.5) + ((1.0 + ((eps * eps) * -0.16666666666666666)) * cos(x)));
end

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\cos x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) + \cos \color{blue}{x}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right), \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\sin x \cdot \frac{-1}{2}\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \sin x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \cos x\right)}\right) + \cos x\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right) + \cos x\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x + \cos \color{blue}{x}\right)\right)\right) \]

   14. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \color{blue}{\cos x}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right), \color{blue}{\cos x}\right)\right)\right) \]

5. Simplified 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \sin x\right) \cdot -0.5 + \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \cdot \cos x\right) \]
7. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}, 2\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right), 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), 2\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), 2\right) \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), 2\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), 2\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), 2\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), 2\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   20. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

7. Simplified 99.8%

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

Alternative 4: 99.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return 2.0 * (cos((x + (eps * 0.5))) * (eps * (0.5 + (eps * (eps * -0.020833333333333332)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * (cos((x + (eps * 0.5d0))) * (eps * (0.5d0 + (eps * (eps * (-0.020833333333333332d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}, 2\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right), 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), 2\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), 2\right) \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), 2\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), 2\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), 2\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), 2\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   20. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2} \cdot \frac{-1}{48}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{48}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{-1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   7. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{-1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{2}\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}, 2\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right), 2\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right), 2\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \varepsilon\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right), 2\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right)\right), 2\right) \]

   9. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + \left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)\right)\right)\right), 2\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)\right)\right), 2\right) \]

   12. distribute-rgt-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(\left(x \cdot 2\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot \left(2 \cdot \frac{1}{2}\right) + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x \cdot 1 + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon \cdot \frac{1}{2}\right)\right)\right), 2\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\left(x + \frac{1}{2} \cdot \varepsilon\right)\right)\right), 2\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \varepsilon\right)\right)\right)\right), 2\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   20. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   2. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 6: 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 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\cos x}\right) \]

   2. cos-lowering-cos.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right) \]

5. Simplified 98.9%

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

Alternative 7: 98.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right) + \varepsilon \cdot \left(x \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    1.0
    (*
     (* x x)
     (+
      -0.5
      (* x (* x (+ 0.041666666666666664 (* (* x x) -0.001388888888888889)))))))
   (* eps (* x -0.5)))))

C

double code(double x, double eps) {
	return eps * ((1.0 + ((x * x) * (-0.5 + (x * (x * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))) + (eps * (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) + (x * (x * (0.041666666666666664d0 + ((x * x) * (-0.001388888888888889d0)))))))) + (eps * (x * (-0.5d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((1.0 + ((x * x) * (-0.5 + (x * (x * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))) + (eps * (x * -0.5)));
}

Python

def code(x, eps):
	return eps * ((1.0 + ((x * x) * (-0.5 + (x * (x * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))) + (eps * (x * -0.5)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(1.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(x * Float64(x * Float64(0.041666666666666664 + Float64(Float64(x * x) * -0.001388888888888889))))))) + Float64(eps * Float64(x * -0.5))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((1.0 + ((x * x) * (-0.5 + (x * (x * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))) + (eps * (x * -0.5)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(x * N[(x * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

8. Simplified 98.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    17. *-lowering-*.f64 97.8%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

11. Simplified 97.8%

    \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} + \varepsilon \cdot \left(x \cdot -0.5\right)\right) \]
12. Add Preprocessing

Alternative 8: 98.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    x
    (+
     (* eps -0.5)
     (*
      x
      (+
       -0.5
       (*
        (* x x)
        (+ 0.041666666666666664 (* (* x x) -0.001388888888888889))))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))));
}

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 * ((-0.5d0) + ((x * x) * (0.041666666666666664d0 + ((x * x) * (-0.001388888888888889d0)))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(Float64(eps * -0.5) + Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * -0.001388888888888889)))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

8. Simplified 98.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    17. *-lowering-*.f64 97.8%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

11. Simplified 97.8%

    \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} + \varepsilon \cdot \left(x \cdot -0.5\right)\right) \]

12. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)} \]
13. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot x + \color{blue}{{x}^{2}} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon\right) + \color{blue}{{x}^{2}} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon\right) + \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon\right) + x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    7. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

    13. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right)\right)\right) \]

14. Simplified 97.8%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \]
15. Add Preprocessing

Alternative 9: 98.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    x
    (*
     x
     (+
      -0.5
      (*
       (* x x)
       (+ 0.041666666666666664 (* (* x x) -0.001388888888888889)))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889)))))));
}

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) + ((x * x) * (0.041666666666666664d0 + ((x * x) * (-0.001388888888888889d0))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889)))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(Float64(x * x) * -0.001388888888888889))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (x * (-0.5 + ((x * x) * (0.041666666666666664 + ((x * x) * -0.001388888888888889)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

8. Simplified 98.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\varepsilon}, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

    17. *-lowering-*.f64 97.8%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right)\right) \]

11. Simplified 97.8%

    \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} + \varepsilon \cdot \left(x \cdot -0.5\right)\right) \]

12. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \]
13. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)} - \frac{1}{2}\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{720}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

    18. *-lowering-*.f64 97.7%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right)\right)\right) \]

14. Simplified 97.7%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \]
15. Add Preprocessing

Alternative 10: 98.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (* x (+ (* eps -0.5) (* x (+ -0.5 (* (* x x) 0.041666666666666664))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * 0.041666666666666664))))));
}

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 * ((-0.5d0) + ((x * x) * 0.041666666666666664d0))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * 0.041666666666666664))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * 0.041666666666666664))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(Float64(eps * -0.5) + Float64(x * Float64(-0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * ((eps * -0.5) + (x * (-0.5 + ((x * x) * 0.041666666666666664))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

8. Simplified 98.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)}\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

    14. *-lowering-*.f64 97.5%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right)\right)\right)\right) \]

11. Simplified 97.5%

    \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)} \]
12. Add Preprocessing

Alternative 11: 98.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* (* x x) (+ -0.5 (* (* x x) 0.041666666666666664))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (-0.5 + ((x * x) * 0.041666666666666664))));
}

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) + ((x * x) * 0.041666666666666664d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (-0.5 + ((x * x) * 0.041666666666666664))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((x * x) * (-0.5 + ((x * x) * 0.041666666666666664))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((x * x) * (-0.5 + ((x * x) * 0.041666666666666664))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\cos x}\right) \]

   2. cos-lowering-cos.f64 98.9%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right) \]

5. Simplified 98.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(1 + \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(1 + x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(1 + \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \cdot x\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 97.5%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right)\right)} \]

9. Final simplification 97.5%

   \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right) \]
10. Add Preprocessing

Alternative 12: 98.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)\right)}\right) \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\varepsilon, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 97.4%

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

Alternative 13: 98.3% 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 62.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\cos x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)}\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\varepsilon \cdot \left(\sin x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon \cdot \sin x\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\sin x}\right)\right)\right)\right) \]

   12. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)\right)}\right) \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

   3. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\varepsilon, \color{blue}{x}\right)\right)\right)\right)\right) \]

8. Simplified 97.4%

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

9. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

    8. *-lowering-*.f64 97.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

11. Simplified 97.4%

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

Alternative 14: 97.8% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* (* eps eps) -0.16666666666666666))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((eps * eps) * (-0.16666666666666666d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666));
}

Python

def code(x, eps):
	return eps * (1.0 + ((eps * eps) * -0.16666666666666666))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * -0.16666666666666666)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((eps * eps) * -0.16666666666666666));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \color{blue}{\cos x}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right) + \cos \color{blue}{x}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x\right)\right), \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\sin x \cdot \frac{-1}{2}\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\varepsilon \cdot \sin x\right)\right), \left(\color{blue}{\varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \sin x\right)\right), \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \cos x\right)\right)\right) \]

   11. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\varepsilon \cdot \cos x\right)}\right) + \cos x\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right) + \cos x\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x + \cos \color{blue}{x}\right)\right)\right) \]

   14. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \color{blue}{\cos x}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right), \color{blue}{\cos x}\right)\right)\right) \]

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({\varepsilon}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 97.0%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right)\right)\right) \]

8. Simplified 97.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

9. Final simplification 97.0%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 15: 97.9% 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 62.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sin \varepsilon} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 97.0%

      \[\leadsto \mathsf{sin.f64}\left(\varepsilon\right) \]

5. Simplified 97.0%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon} \]
7. Step-by-step derivation

8. Simplified 97.0%

   \[\leadsto \color{blue}{\varepsilon} \]
9. 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 2024150 
(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)))