2sin (example 3.3)

Percentage Accurate: 62.1% → 99.9%

Time: 15.9s

Alternatives: 14

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\cos \left(x + \varepsilon \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.00026041666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot -1.5500992063492063 \cdot 10^{-6}\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (cos (+ x (* eps 0.5)))
   (*
    eps
    (+
     0.5
     (*
      (* eps eps)
      (+
       -0.020833333333333332
       (*
        (* eps eps)
        (+
         0.00026041666666666666
         (* (* eps eps) -1.5500992063492063e-6))))))))))

C

double code(double x, double eps) {
	return 2.0 * (cos((x + (eps * 0.5))) * (eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * (cos((x + (eps * 0.5d0))) * (eps * (0.5d0 + ((eps * eps) * ((-0.020833333333333332d0) + ((eps * eps) * (0.00026041666666666666d0 + ((eps * eps) * (-1.5500992063492063d-6)))))))))
end function

Java

public static double code(double x, double eps) {
	return 2.0 * (Math.cos((x + (eps * 0.5))) * (eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6))))))));
}

Python

def code(x, eps):
	return 2.0 * (math.cos((x + (eps * 0.5))) * (eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6))))))))

Julia

function code(x, eps)
	return Float64(2.0 * Float64(cos(Float64(x + Float64(eps * 0.5))) * Float64(eps * Float64(0.5 + Float64(Float64(eps * eps) * Float64(-0.020833333333333332 + Float64(Float64(eps * eps) * Float64(0.00026041666666666666 + Float64(Float64(eps * eps) * -1.5500992063492063e-6)))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * (cos((x + (eps * 0.5))) * (eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6))))))));
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(x + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(0.5 + N[(N[(eps * eps), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(eps * eps), $MachinePrecision] * N[(0.00026041666666666666 + N[(N[(eps * eps), $MachinePrecision] * -1.5500992063492063e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \frac{-1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-1}{48} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3840}, \left(\frac{-1}{645120} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3840}, \left({\varepsilon}^{2} \cdot \frac{-1}{645120}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3840}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{-1}{645120}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3840}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-1}{645120}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{1}{3840}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{645120}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right)\right)\right)\right), 2\right) \]

10. Simplified 99.8%

    \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.00026041666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot -1.5500992063492063 \cdot 10^{-6}\right)\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right) \cdot 2 \]

11. Final simplification 99.8%

    \[\leadsto 2 \cdot \left(\cos \left(x + \varepsilon \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.00026041666666666666 + \left(\varepsilon \cdot \varepsilon\right) \cdot -1.5500992063492063 \cdot 10^{-6}\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \varepsilon \cdot \left(\varepsilon \cdot 0.00026041666666666666\right)\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (+
    0.5
    (*
     (* eps eps)
     (+ -0.020833333333333332 (* eps (* eps 0.00026041666666666666)))))
   (* eps (cos (+ x (* eps 0.5)))))))

C

double code(double x, double eps) {
	return 2.0 * ((0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666))))) * (eps * cos((x + (eps * 0.5)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * ((0.5d0 + ((eps * eps) * ((-0.020833333333333332d0) + (eps * (eps * 0.00026041666666666666d0))))) * (eps * cos((x + (eps * 0.5d0)))))
end function

Java

public static double code(double x, double eps) {
	return 2.0 * ((0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666))))) * (eps * Math.cos((x + (eps * 0.5)))));
}

Python

def code(x, eps):
	return 2.0 * ((0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666))))) * (eps * math.cos((x + (eps * 0.5)))))

Julia

function code(x, eps)
	return Float64(2.0 * Float64(Float64(0.5 + Float64(Float64(eps * eps) * Float64(-0.020833333333333332 + Float64(eps * Float64(eps * 0.00026041666666666666))))) * Float64(eps * cos(Float64(x + Float64(eps * 0.5))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * ((0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666))))) * (eps * cos((x + (eps * 0.5)))));
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[(0.5 + N[(N[(eps * eps), $MachinePrecision] * N[(-0.020833333333333332 + N[(eps * N[(eps * 0.00026041666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[Cos[N[(x + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \frac{-1}{48}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2} \cdot \frac{1}{3840}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3840}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

   14. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{3840}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), 2\right) \]

7. Simplified 99.6%

   \[\leadsto \left(\color{blue}{\left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \varepsilon \cdot \left(\varepsilon \cdot 0.00026041666666666666\right)\right)\right)\right)} \cdot \cos \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot 2 \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)\right)}, 2\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right) \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), 2\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \frac{-1}{48}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \left(\frac{-1}{48} + \frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\frac{1}{3840} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2} \cdot \frac{1}{3840}\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3840}\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left(\varepsilon \cdot \left(\frac{1}{3840} \cdot \varepsilon\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{3840} \cdot \varepsilon\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{3840}\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{3840}\right)\right)\right)\right)\right), \left(\varepsilon \cdot \cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)\right), 2\right) \]

10. Simplified 99.6%

    \[\leadsto \color{blue}{\left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \varepsilon \cdot \left(\varepsilon \cdot 0.00026041666666666666\right)\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)\right)} \cdot 2 \]

11. Final simplification 99.6%

    \[\leadsto 2 \cdot \left(\left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.020833333333333332 + \varepsilon \cdot \left(\varepsilon \cdot 0.00026041666666666666\right)\right)\right) \cdot \left(\varepsilon \cdot \cos \left(x + \varepsilon \cdot 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 4: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

   6. clear-num N/A

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

   7. sin-cos-mult N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 99.0%

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

10. Simplified 99.0%

    \[\leadsto \frac{2}{\frac{1}{\color{blue}{\left(\varepsilon \cdot \left(0.5 + -0.020833333333333332 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot 0.5\right)}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{2}{1} \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right)} \]

    2. metadata-eval N/A

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{2} \]

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. associate-*l* N/A

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot 2\right)} \]

    7. associate-*r* N/A

       \[\leadsto \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\cos \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot 2\right)}\right) \]

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

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

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

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

    10. associate-*r* N/A

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

    11. *-commutative N/A

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

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

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

    13. *-commutative N/A

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

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

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

    15. *-commutative N/A

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

    16. associate-*r* N/A

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

    17. *-commutative N/A

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 5: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

   6. clear-num N/A

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

   7. sin-cos-mult N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 99.0%

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

10. Simplified 99.0%

    \[\leadsto \frac{2}{\frac{1}{\color{blue}{\left(\varepsilon \cdot \left(0.5 + -0.020833333333333332 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot 0.5\right)}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{2}{1} \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right)} \]

    2. metadata-eval N/A

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto 2 \cdot \left(\cos \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(2 \cdot \cos \left(x + \varepsilon \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]

    5. *-commutative N/A

       \[\leadsto \left(\cos \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot 2\right) \cdot \left(\color{blue}{\varepsilon} \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \left(\cos \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot 2\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\varepsilon}\right) \]

    7. associate-*r* N/A

       \[\leadsto \left(\left(\cos \left(x + \varepsilon \cdot \frac{1}{2}\right) \cdot 2\right) \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

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

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 6: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 7: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \frac{1}{\frac{1}{\cos \left(x + \varepsilon \cdot 0.5\right)}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* 2.0 (* (* eps 0.5) (/ 1.0 (/ 1.0 (cos (+ x (* eps 0.5))))))))

C

double code(double x, double eps) {
	return 2.0 * ((eps * 0.5) * (1.0 / (1.0 / cos((x + (eps * 0.5))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * ((eps * 0.5d0) * (1.0d0 / (1.0d0 / cos((x + (eps * 0.5d0))))))
end function

Java

public static double code(double x, double eps) {
	return 2.0 * ((eps * 0.5) * (1.0 / (1.0 / Math.cos((x + (eps * 0.5))))));
}

Python

def code(x, eps):
	return 2.0 * ((eps * 0.5) * (1.0 / (1.0 / math.cos((x + (eps * 0.5))))))

Julia

function code(x, eps)
	return Float64(2.0 * Float64(Float64(eps * 0.5) * Float64(1.0 / Float64(1.0 / cos(Float64(x + Float64(eps * 0.5)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * ((eps * 0.5) * (1.0 / (1.0 / cos((x + (eps * 0.5))))));
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[(eps * 0.5), $MachinePrecision] * N[(1.0 / N[(1.0 / N[Cos[N[(x + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.8%

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

10. Simplified 98.8%

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

    1. remove-double-div N/A

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

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

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

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

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

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

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

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

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

    6. *-lowering-*.f64 98.8%

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

12. Applied egg-rr 98.8%

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

13. Final simplification 98.8%

    \[\leadsto 2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \frac{1}{\frac{1}{\cos \left(x + \varepsilon \cdot 0.5\right)}}\right) \]
14. Add Preprocessing

Alternative 8: 99.6% 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 63.8%

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

   1. diff-sin N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. *-rgt-identity N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.8%

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

10. Simplified 98.8%

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

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. associate-*l* N/A

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

    4. metadata-eval N/A

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

    5. *-rgt-identity N/A

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

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

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

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

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

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

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

    9. *-lowering-*.f64 98.8%

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

12. Applied egg-rr 98.8%

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

13. Final simplification 98.8%

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

Alternative 9: 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 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

Alternative 10: 98.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(\left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right) + x \cdot \left(\varepsilon \cdot 0.08333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+ 1.0 (* (* eps eps) -0.16666666666666666))
   (*
    x
    (+
     (* eps -0.5)
     (*
      x
      (+
       (+ -0.5 (* (* eps eps) 0.08333333333333333))
       (* x (* eps 0.08333333333333333)))))))))

C

double code(double x, double eps) {
	return eps * ((1.0 + ((eps * eps) * -0.16666666666666666)) + (x * ((eps * -0.5) + (x * ((-0.5 + ((eps * eps) * 0.08333333333333333)) + (x * (eps * 0.08333333333333333)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((1.0d0 + ((eps * eps) * (-0.16666666666666666d0))) + (x * ((eps * (-0.5d0)) + (x * (((-0.5d0) + ((eps * eps) * 0.08333333333333333d0)) + (x * (eps * 0.08333333333333333d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((1.0 + ((eps * eps) * -0.16666666666666666)) + (x * ((eps * -0.5) + (x * ((-0.5 + ((eps * eps) * 0.08333333333333333)) + (x * (eps * 0.08333333333333333)))))));
}

Python

def code(x, eps):
	return eps * ((1.0 + ((eps * eps) * -0.16666666666666666)) + (x * ((eps * -0.5) + (x * ((-0.5 + ((eps * eps) * 0.08333333333333333)) + (x * (eps * 0.08333333333333333)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(1.0 + Float64(Float64(eps * eps) * -0.16666666666666666)) + Float64(x * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(-0.5 + Float64(Float64(eps * eps) * 0.08333333333333333)) + Float64(x * Float64(eps * 0.08333333333333333))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((1.0 + ((eps * eps) * -0.16666666666666666)) + (x * ((eps * -0.5) + (x * ((-0.5 + ((eps * eps) * 0.08333333333333333)) + (x * (eps * 0.08333333333333333)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(-0.5 + N[(N[(eps * eps), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft1-in N/A

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

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

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left({\varepsilon}^{2} \cdot \frac{-1}{6}\right)\right), \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{-1}{6}\right)\right), \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-1}{6}\right)\right), \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{6}\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right), \color{blue}{\left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)}\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(\left(-0.5 + 0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon \cdot 0.08333333333333333\right)\right)\right)\right)} \]

9. Final simplification 97.5%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + x \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(\left(-0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333\right) + x \cdot \left(\varepsilon \cdot 0.08333333333333333\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 98.3% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft1-in N/A

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

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

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. distribute-lft-out N/A

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

   9. associate-*r* N/A

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

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

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

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

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

   12. unpow2 N/A

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

   13. distribute-lft-out N/A

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

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

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

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

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 12: 98.3% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft1-in N/A

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

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

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. distribute-lft-out N/A

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

   9. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 97.5%

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

8. Simplified 97.5%

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

Alternative 13: 97.8% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in x around 0

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

   1. sin-lowering-sin.f64 97.0%

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

5. Simplified 97.0%

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

6. Taylor expanded in eps around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 97.0%

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

8. Simplified 97.0%

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

Alternative 14: 97.8% 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 63.8%

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

3. Taylor expanded in x around 0

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

   1. sin-lowering-sin.f64 97.0%

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

5. Simplified 97.0%

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

6. Taylor expanded in eps around 0

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

8. Simplified 97.0%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024164 
(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)))