2sin (example 3.3)

Percentage Accurate: 62.6% → 99.7%

Time: 18.1s

Alternatives: 14

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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)))) * cos(x)) + ((eps * sin(x)) * ((-0.5d0) + (eps * (eps * 0.041666666666666664d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (((1.0 + (eps * (eps * -0.16666666666666666))) * Math.cos(x)) + ((eps * Math.sin(x)) * (-0.5 + (eps * (eps * 0.041666666666666664)))));
}

Python

def code(x, eps):
	return eps * (((1.0 + (eps * (eps * -0.16666666666666666))) * math.cos(x)) + ((eps * math.sin(x)) * (-0.5 + (eps * (eps * 0.041666666666666664)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((1.0 + (eps * (eps * -0.16666666666666666))) * cos(x)) + ((eps * sin(x)) * (-0.5 + (eps * (eps * 0.041666666666666664)))));
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666\right)\right) \cdot \cos x + \left(\varepsilon \cdot \sin x\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. distribute-lft1-in N/A

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

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

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.9%

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

5. Final simplification 99.9%

   \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \]
6. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.6%

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

7. Applied egg-rr 99.6%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 6: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \cos x\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * cos(x)) * (1.0d0 + ((-0.16666666666666666d0) * (eps * eps)))) + (x * ((eps * eps) * ((-0.5d0) + (eps * (eps * 0.041666666666666664d0)))))
end function

Java

public static double code(double x, double eps) {
	return ((eps * Math.cos(x)) * (1.0 + (-0.16666666666666666 * (eps * eps)))) + (x * ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))));
}

Python

def code(x, eps):
	return ((eps * math.cos(x)) * (1.0 + (-0.16666666666666666 * (eps * eps)))) + (x * ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * cos(x)) * (1.0 + (-0.16666666666666666 * (eps * eps)))) + (x * ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))));
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

5. Simplified 100.0%

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

   1. distribute-rgt-in N/A

      \[\leadsto \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right) \cdot \cos x\right) \cdot \varepsilon + \color{blue}{\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right) \cdot \cos x\right) \cdot \varepsilon\right), \color{blue}{\left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)}\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right) \cdot \left(\cos x \cdot \varepsilon\right)\right), \left(\color{blue}{\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right)} \cdot \varepsilon\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right) \cdot \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)}\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\color{blue}{\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right)} \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{6}\right)\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\color{blue}{\left(\varepsilon \cdot \sin x\right)} \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6}\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \color{blue}{\sin x}\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \color{blue}{\sin x}\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \color{blue}{\sin x}\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(\varepsilon \cdot \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)\right) \cdot \varepsilon\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \cos x\right)\right), \left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)}\right) \cdot \varepsilon\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\left(\varepsilon \cdot \sin x\right) \cdot \left(\frac{-1}{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)}\right)\right) \cdot \varepsilon\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \sin x\right) \cdot \color{blue}{\left(\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right) \cdot \varepsilon\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\left(\varepsilon \cdot \sin x\right), \color{blue}{\left(\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right) \cdot \varepsilon\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \sin x\right), \left(\color{blue}{\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right)} \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right), \left(\left(\frac{-1}{2} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)}\right) \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{sin.f64}\left(x\right)\right), \mathsf{*.f64}\left(\left(\frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right), \color{blue}{\varepsilon}\right)\right)\right) \]

7. Applied egg-rr 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \color{blue}{\left({\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left({\varepsilon}^{2} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(x \cdot {\varepsilon}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {\varepsilon}^{2}} - \frac{1}{2}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \left(x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\color{blue}{\frac{1}{24} \cdot {\varepsilon}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\color{blue}{\frac{1}{24} \cdot {\varepsilon}^{2}} - \frac{1}{2}\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{24} \cdot {\varepsilon}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{24} \cdot {\varepsilon}^{2} + \frac{-1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(\frac{1}{24} \cdot {\varepsilon}^{2}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left({\varepsilon}^{2} \cdot \frac{1}{24}\right), \frac{-1}{2}\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24}\right), \frac{-1}{2}\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{24} \cdot \varepsilon\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{24} \cdot \varepsilon\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{24}\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\varepsilon, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{24}\right)\right), \frac{-1}{2}\right)\right)\right)\right) \]

10. Simplified 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \left(\varepsilon \cdot \cos x\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right)\right) \]
12. Add Preprocessing

Alternative 7: 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 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.1%

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

5. Simplified 99.1%

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

Alternative 8: 98.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \varepsilon \cdot t\_0 + \left(x \cdot -0.5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.16666666666666666 (* eps eps)))))
   (+ (* eps t_0) (* (* x -0.5) (* eps (+ eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (-0.16666666666666666 * (eps * eps));
	return (eps * t_0) + ((x * -0.5) * (eps * (eps + (x * t_0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = 1.0d0 + ((-0.16666666666666666d0) * (eps * eps))
    code = (eps * t_0) + ((x * (-0.5d0)) * (eps * (eps + (x * t_0))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (-0.16666666666666666 * (eps * eps));
	return (eps * t_0) + ((x * -0.5) * (eps * (eps + (x * t_0))));
}

Python

def code(x, eps):
	t_0 = 1.0 + (-0.16666666666666666 * (eps * eps))
	return (eps * t_0) + ((x * -0.5) * (eps * (eps + (x * t_0))))

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(-0.16666666666666666 * Float64(eps * eps)))
	return Float64(Float64(eps * t_0) + Float64(Float64(x * -0.5) * Float64(eps * Float64(eps + Float64(x * t_0)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = 1.0 + (-0.16666666666666666 * (eps * eps));
	tmp = (eps * t_0) + ((x * -0.5) * (eps * (eps + (x * t_0))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * t$95$0), $MachinePrecision] + N[(N[(x * -0.5), $MachinePrecision] * N[(eps * N[(eps + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

5. Simplified 100.0%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
7. 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

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. associate-*l* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-+r+ N/A

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

    5. unpow2 N/A

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

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

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

    7. distribute-lft-out N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

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

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

    11. *-commutative N/A

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

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

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

    13. unpow2 N/A

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

    14. distribute-lft-out N/A

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

    15. +-commutative N/A

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

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

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

11. Simplified 98.5%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(x \cdot -0.5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon + x \cdot \left(1 + -0.16666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 9: 98.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (((1.0 + (eps * (eps * -0.16666666666666666))) * (1.0 + (-0.5 * (x * x)))) + (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 + (eps * (eps * (-0.16666666666666666d0)))) * (1.0d0 + ((-0.5d0) * (x * x)))) + (eps * (x * (-0.5d0))))
end function

Java

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

Python

def code(x, eps):
	return eps * (((1.0 + (eps * (eps * -0.16666666666666666))) * (1.0 + (-0.5 * (x * x)))) + (eps * (x * -0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. metadata-eval N/A

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

5. Simplified 98.4%

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

6. Taylor expanded in eps around 0

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

8. Simplified 98.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666\right)\right) \cdot \left(1 + -0.5 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(x \cdot -0.5\right)\right)} \]
9. Add Preprocessing

Alternative 10: 98.4% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + x \cdot \left(\frac{-1}{2} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right)}\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot {\varepsilon}^{2} + \left(\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot x + \color{blue}{\left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x}\right)\right)\right)\right) \]

   4. associate-+r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{2} \cdot {\varepsilon}^{2} + \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot x\right) + \color{blue}{\left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot x + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + \color{blue}{\left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)} \cdot x\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + \left(\color{blue}{\frac{1}{12}} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}\right) + x \cdot \color{blue}{\left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\varepsilon + x \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \left(\varepsilon + x\right)\right) + x \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot x\right)\right) \cdot 0.08333333333333333\right)\right)} \]
9. Add Preprocessing

Alternative 11: 98.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}\right)\right)}\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) + \frac{-1}{2}\right)\right)\right)\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + \left(\left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right) \cdot x + \color{blue}{\frac{-1}{2} \cdot x}\right)\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + \left(\left(\left(\frac{1}{12} \cdot \varepsilon\right) \cdot x\right) \cdot x + \frac{-1}{2} \cdot x\right)\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + \left(\left(\frac{1}{12} \cdot \varepsilon\right) \cdot \left(x \cdot x\right) + \color{blue}{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + \left(\left(\frac{1}{12} \cdot \varepsilon\right) \cdot {x}^{2} + \frac{-1}{2} \cdot x\right)\right)\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} \cdot \varepsilon + \left(\frac{1}{12} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}} \cdot x\right)\right)\right)\right)\right) \]

   10. associate-+r+ N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{12} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{2} \cdot x}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{12} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right)\right) \]

8. Simplified 98.2%

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

9. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

    3. distribute-lft-out N/A

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

    4. associate-*l* N/A

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

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

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

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

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

    7. unpow2 N/A

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

    8. distribute-lft-out N/A

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

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

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

    10. +-lowering-+.f64 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

    \[\leadsto \varepsilon + -0.5 \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\right) \]
13. 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 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

      \[\leadsto \left(\varepsilon + \frac{-1}{2} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) + \left(\frac{-1}{2} \cdot {\varepsilon}^{2}\right) \cdot x \]

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-rgt1-in N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. associate-*r* N/A

       \[\leadsto \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \cdot \varepsilon + \left(\frac{-1}{2} \cdot \left(x \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   16. *-commutative N/A

       \[\leadsto \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \cdot \varepsilon + \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)\right) \cdot \varepsilon \]

   17. associate-*r* N/A

       \[\leadsto \left(1 + \frac{-1}{2} \cdot {x}^{2}\right) \cdot \varepsilon + \left(\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot x\right) \cdot \varepsilon \]

   18. distribute-rgt-out N/A

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

   19. associate-+r+ N/A

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

8. Simplified 98.2%

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

Alternative 13: 98.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. metadata-eval N/A

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

5. Simplified 98.4%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]
7. 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. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 98.1%

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

8. Simplified 98.1%

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

Alternative 14: 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 61.0%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 97.6%

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

5. Simplified 97.6%

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

6. Taylor expanded in eps around 0

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

8. Simplified 97.6%

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024185 
(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))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

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