2sin (example 3.3)

Percentage Accurate: 62.6% → 99.5%

Time: 13.7s

Alternatives: 13

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sin-lowering-sin.f64 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 100.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 100.0%

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

10. Simplified 100.0%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

Alternative 4: 98.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (x * (eps * (x * ((-0.5d0) + (x * (x * (0.041666666666666664d0 + (x * (x * (-0.001388888888888889d0))))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.0%

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

8. Simplified 99.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right) + \color{blue}{1}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(\frac{-1}{2} + \left(x \cdot x\right) \cdot \left(\frac{1}{24} + \left(x \cdot x\right) \cdot \frac{-1}{720}\right)\right)\right)\right) \cdot \varepsilon + \color{blue}{1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

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

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

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

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 5: 98.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * ((x * x) * ((-0.5d0) + ((0.041666666666666664d0 + (x * (x * (-0.001388888888888889d0)))) * (x * x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.0%

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

8. Simplified 99.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-*l* N/A

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

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

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

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

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

   12. *-lowering-*.f64 99.0%

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

10. Applied egg-rr 99.0%

    \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\left(-0.5 + x \cdot \left(x \cdot \left(0.041666666666666664 + x \cdot \left(x \cdot -0.001388888888888889\right)\right)\right)\right) \cdot \left(x \cdot x\right)}\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \left(\left(\frac{-1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{-1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{1}\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \left(\left(\frac{-1}{2} + x \cdot \left(x \cdot \left(\frac{1}{24} + x \cdot \left(x \cdot \frac{-1}{720}\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon + \color{blue}{1 \cdot \varepsilon} \]

    3. *-lft-identity N/A

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

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

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

12. Applied egg-rr 99.0%

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

13. Final simplification 99.0%

    \[\leadsto \varepsilon + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(-0.5 + \left(0.041666666666666664 + x \cdot \left(x \cdot -0.001388888888888889\right)\right) \cdot \left(x \cdot x\right)\right)\right) \]
14. Add Preprocessing

Alternative 6: 98.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.0%

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

8. Simplified 99.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + \left(x \cdot x\right) \cdot -0.001388888888888889\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-*l* N/A

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

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

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

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

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

   12. *-lowering-*.f64 99.0%

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

10. Applied egg-rr 99.0%

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

Alternative 7: 98.5% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (x * ((-0.5d0) + ((x * x) * (0.041666666666666664d0 + ((-0.001388888888888889d0) * (x * x))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 98.5% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 9: 98.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

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))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sin-lowering-sin.f64 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-out N/A

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

   11. +-commutative N/A

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

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

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

   13. distribute-lft-out N/A

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

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

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

   15. +-lowering-+.f64 98.9%

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

8. Simplified 98.9%

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

Alternative 10: 98.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sin-lowering-sin.f64 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. +-lowering-+.f64 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 11: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sin-lowering-sin.f64 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-out N/A

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

   11. +-commutative N/A

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

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

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

   13. distribute-lft-out N/A

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

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

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

   15. +-lowering-+.f64 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 98.8%

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

11. Simplified 98.8%

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

Alternative 12: 98.4% 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 64.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

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

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 98.8%

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

8. Simplified 98.8%

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

Alternative 13: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

   12. sin-lowering-sin.f64 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 98.4%

   \[\leadsto \color{blue}{\varepsilon} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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