2cos (problem 3.3.5)

Percentage Accurate: 51.3% → 99.7%

Time: 18.6s

Alternatives: 16

Speedup: 41.0×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf

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

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

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

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

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf

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

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

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

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

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

   3. metadata-eval N/A

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

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

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

   14. *-commutative N/A

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

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

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

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

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return -2.0 * (eps * (sin((x + (eps * 0.5))) * (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) * (eps * (sin((x + (eps * 0.5d0))) * (0.5d0 + ((eps * eps) * (-0.020833333333333332d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf

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

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

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

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

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

10. Simplified 99.4%

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

11. Taylor expanded in eps around 0

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

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

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 99.2%

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

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

Alternative 5: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 6: 98.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(\varepsilon \cdot \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* (* eps eps) (+ -0.5 (* eps (* eps 0.041666666666666664))))
  (*
   x
   (+
    (* eps (+ -1.0 (* eps (* eps 0.16666666666666666))))
    (*
     x
     (+
      (*
       x
       (* eps (+ 0.16666666666666666 (* eps (* eps -0.027777777777777776)))))
      (* eps (* eps (+ 0.25 (* eps (* eps -0.020833333333333332)))))))))))

C

double code(double x, double eps) {
	return ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((eps * (-1.0 + (eps * (eps * 0.16666666666666666)))) + (x * ((x * (eps * (0.16666666666666666 + (eps * (eps * -0.027777777777777776))))) + (eps * (eps * (0.25 + (eps * (eps * -0.020833333333333332)))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * eps) * ((-0.5d0) + (eps * (eps * 0.041666666666666664d0)))) + (x * ((eps * ((-1.0d0) + (eps * (eps * 0.16666666666666666d0)))) + (x * ((x * (eps * (0.16666666666666666d0 + (eps * (eps * (-0.027777777777777776d0)))))) + (eps * (eps * (0.25d0 + (eps * (eps * (-0.020833333333333332d0))))))))))
end function

Java

public static double code(double x, double eps) {
	return ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((eps * (-1.0 + (eps * (eps * 0.16666666666666666)))) + (x * ((x * (eps * (0.16666666666666666 + (eps * (eps * -0.027777777777777776))))) + (eps * (eps * (0.25 + (eps * (eps * -0.020833333333333332)))))))));
}

Python

def code(x, eps):
	return ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((eps * (-1.0 + (eps * (eps * 0.16666666666666666)))) + (x * ((x * (eps * (0.16666666666666666 + (eps * (eps * -0.027777777777777776))))) + (eps * (eps * (0.25 + (eps * (eps * -0.020833333333333332)))))))))

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * eps) * Float64(-0.5 + Float64(eps * Float64(eps * 0.041666666666666664)))) + Float64(x * Float64(Float64(eps * Float64(-1.0 + Float64(eps * Float64(eps * 0.16666666666666666)))) + Float64(x * Float64(Float64(x * Float64(eps * Float64(0.16666666666666666 + Float64(eps * Float64(eps * -0.027777777777777776))))) + Float64(eps * Float64(eps * Float64(0.25 + Float64(eps * Float64(eps * -0.020833333333333332))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * eps) * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((eps * (-1.0 + (eps * (eps * 0.16666666666666666)))) + (x * ((x * (eps * (0.16666666666666666 + (eps * (eps * -0.027777777777777776))))) + (eps * (eps * (0.25 + (eps * (eps * -0.020833333333333332)))))))));
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * N[(-0.5 + N[(eps * N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(-1.0 + N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(x * N[(eps * N[(0.16666666666666666 + N[(eps * N[(eps * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(0.25 + N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(\varepsilon \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)} \]

8. Simplified 97.0%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(\varepsilon \cdot \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 7: 98.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(x \cdot \left(-1 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(0.25 \cdot \left(x \cdot x\right) + \left(-0.5 + \varepsilon \cdot \left(x \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* x (+ -1.0 (* 0.16666666666666666 (* x x))))
   (*
    eps
    (+
     (* 0.25 (* x x))
     (+
      -0.5
      (*
       eps
       (+
        (* x (+ 0.16666666666666666 (* -0.027777777777777776 (* x x))))
        (*
         eps
         (+ 0.041666666666666664 (* -0.020833333333333332 (* x x))))))))))))

C

double code(double x, double eps) {
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * ((0.25 * (x * x)) + (-0.5 + (eps * ((x * (0.16666666666666666 + (-0.027777777777777776 * (x * x)))) + (eps * (0.041666666666666664 + (-0.020833333333333332 * (x * x))))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((x * ((-1.0d0) + (0.16666666666666666d0 * (x * x)))) + (eps * ((0.25d0 * (x * x)) + ((-0.5d0) + (eps * ((x * (0.16666666666666666d0 + ((-0.027777777777777776d0) * (x * x)))) + (eps * (0.041666666666666664d0 + ((-0.020833333333333332d0) * (x * x))))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * ((0.25 * (x * x)) + (-0.5 + (eps * ((x * (0.16666666666666666 + (-0.027777777777777776 * (x * x)))) + (eps * (0.041666666666666664 + (-0.020833333333333332 * (x * x))))))))));
}

Python

def code(x, eps):
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * ((0.25 * (x * x)) + (-0.5 + (eps * ((x * (0.16666666666666666 + (-0.027777777777777776 * (x * x)))) + (eps * (0.041666666666666664 + (-0.020833333333333332 * (x * x))))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(x * Float64(-1.0 + Float64(0.16666666666666666 * Float64(x * x)))) + Float64(eps * Float64(Float64(0.25 * Float64(x * x)) + Float64(-0.5 + Float64(eps * Float64(Float64(x * Float64(0.16666666666666666 + Float64(-0.027777777777777776 * Float64(x * x)))) + Float64(eps * Float64(0.041666666666666664 + Float64(-0.020833333333333332 * Float64(x * x)))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * ((0.25 * (x * x)) + (-0.5 + (eps * ((x * (0.16666666666666666 + (-0.027777777777777776 * (x * x)))) + (eps * (0.041666666666666664 + (-0.020833333333333332 * (x * x))))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * N[(-1.0 + N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 + N[(eps * N[(N[(x * N[(0.16666666666666666 + N[(-0.027777777777777776 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(0.041666666666666664 + N[(-0.020833333333333332 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)} \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)}\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right) + \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right), \color{blue}{\left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + -1\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot {x}^{2}\right), -1\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{2}\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot x\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{4} \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{24} + \frac{-1}{48} \cdot {x}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \]

11. Simplified 97.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right) + -1\right) + \varepsilon \cdot \left(0.25 \cdot \left(x \cdot x\right) + \left(\varepsilon \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.027777777777777776\right) + \varepsilon \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(x \cdot x\right)\right)\right) + -0.5\right)\right)\right)} \]

12. Final simplification 97.0%

    \[\leadsto \varepsilon \cdot \left(x \cdot \left(-1 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(0.25 \cdot \left(x \cdot x\right) + \left(-0.5 + \varepsilon \cdot \left(x \cdot \left(0.16666666666666666 + -0.027777777777777776 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(0.041666666666666664 + -0.020833333333333332 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 98.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(\left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right) + x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps (+ -0.5 (* eps (* eps 0.041666666666666664))))
   (*
    x
    (+
     (+ -1.0 (* eps (* eps 0.16666666666666666)))
     (*
      x
      (+
       (* eps (+ 0.25 (* eps (* eps -0.020833333333333332))))
       (*
        x
        (+ 0.16666666666666666 (* eps (* eps -0.027777777777777776)))))))))))

C

double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((-1.0 + (eps * (eps * 0.16666666666666666))) + (x * ((eps * (0.25 + (eps * (eps * -0.020833333333333332)))) + (x * (0.16666666666666666 + (eps * (eps * -0.027777777777777776)))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((-0.5d0) + (eps * (eps * 0.041666666666666664d0)))) + (x * (((-1.0d0) + (eps * (eps * 0.16666666666666666d0))) + (x * ((eps * (0.25d0 + (eps * (eps * (-0.020833333333333332d0))))) + (x * (0.16666666666666666d0 + (eps * (eps * (-0.027777777777777776d0))))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((-1.0 + (eps * (eps * 0.16666666666666666))) + (x * ((eps * (0.25 + (eps * (eps * -0.020833333333333332)))) + (x * (0.16666666666666666 + (eps * (eps * -0.027777777777777776)))))))));
}

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((-1.0 + (eps * (eps * 0.16666666666666666))) + (x * ((eps * (0.25 + (eps * (eps * -0.020833333333333332)))) + (x * (0.16666666666666666 + (eps * (eps * -0.027777777777777776)))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(-0.5 + Float64(eps * Float64(eps * 0.041666666666666664)))) + Float64(x * Float64(Float64(-1.0 + Float64(eps * Float64(eps * 0.16666666666666666))) + Float64(x * Float64(Float64(eps * Float64(0.25 + Float64(eps * Float64(eps * -0.020833333333333332)))) + Float64(x * Float64(0.16666666666666666 + Float64(eps * Float64(eps * -0.027777777777777776))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + (eps * (eps * 0.041666666666666664)))) + (x * ((-1.0 + (eps * (eps * 0.16666666666666666))) + (x * ((eps * (0.25 + (eps * (eps * -0.020833333333333332)))) + (x * (0.16666666666666666 + (eps * (eps * -0.027777777777777776)))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(-0.5 + N[(eps * N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 + N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * N[(0.25 + N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.16666666666666666 + N[(eps * N[(eps * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Final simplification 97.0%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(\left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right) + x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 98.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(x \cdot \left(-1 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(-0.5 + 0.25 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* x (+ -1.0 (* 0.16666666666666666 (* x x))))
   (* eps (+ -0.5 (* 0.25 (* x x)))))))

C

double code(double x, double eps) {
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * (-0.5 + (0.25 * (x * x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((x * ((-1.0d0) + (0.16666666666666666d0 * (x * x)))) + (eps * ((-0.5d0) + (0.25d0 * (x * x)))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * (-0.5 + (0.25 * (x * x)))));
}

Python

def code(x, eps):
	return eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * (-0.5 + (0.25 * (x * x)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(x * Float64(-1.0 + Float64(0.16666666666666666 * Float64(x * x)))) + Float64(eps * Float64(-0.5 + Float64(0.25 * Float64(x * x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((x * (-1.0 + (0.16666666666666666 * (x * x)))) + (eps * (-0.5 + (0.25 * (x * x)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * N[(-1.0 + N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(-0.5 + N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in eps around 0

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

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)}\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right) + \color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right), \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right), \left(\color{blue}{\varepsilon} \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {x}^{2} + -1\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot {x}^{2}\right), -1\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left({x}^{2}\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot x\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right)\right) \]

    12. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\left(-1 \cdot \frac{-1}{4}\right) \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    14. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(-1 \cdot \left(\frac{-1}{4} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

    15. mul-1-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\left(\mathsf{neg}\left(\frac{-1}{4} \cdot {x}^{2}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

    16. distribute-neg-in N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\mathsf{neg}\left(\left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

    17. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{-1}{4} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

    18. mul-1-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(-1 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

    19. distribute-lft-in N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(-1 \cdot \frac{1}{2} + \color{blue}{-1 \cdot \left(\frac{-1}{4} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

    20. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + \color{blue}{-1} \cdot \left(\frac{-1}{4} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

    21. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, x\right)\right), -1\right)\right), \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + \left(-1 \cdot \frac{-1}{4}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right) \]

11. Simplified 97.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right) + -1\right) + \varepsilon \cdot \left(-0.5 + 0.25 \cdot \left(x \cdot x\right)\right)\right)} \]

12. Final simplification 97.0%

    \[\leadsto \varepsilon \cdot \left(x \cdot \left(-1 + 0.16666666666666666 \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot \left(-0.5 + 0.25 \cdot \left(x \cdot x\right)\right)\right) \]
13. Add Preprocessing

Alternative 10: 97.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

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

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

    13. unpow2 N/A

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

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

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

    15. *-commutative N/A

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

    16. associate-*r* N/A

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

    17. *-commutative N/A

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

    18. associate-*r* N/A

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

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

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

11. Simplified 96.7%

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

12. Taylor expanded in eps around 0

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

14. Simplified 96.7%

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

15. Final simplification 96.7%

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

Alternative 11: 97.7% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

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

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

    13. unpow2 N/A

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

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

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

    15. *-commutative N/A

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

    16. associate-*r* N/A

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

    17. *-commutative N/A

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

    18. associate-*r* N/A

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

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

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

11. Simplified 96.7%

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

12. Taylor expanded in eps around 0

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

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

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. unsub-neg N/A

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

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

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

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

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

    10. *-commutative N/A

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

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

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

    12. *-lowering-*.f64 96.6%

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

14. Simplified 96.6%

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

15. Final simplification 96.6%

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

Alternative 12: 53.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.2e-141) (* 0.5 (* x x)) (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-141) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.2d-141)) then
        tmp = 0.5d0 * (x * x)
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.2e-141) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.2e-141:
		tmp = 0.5 * (x * x)
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.2e-141)
		tmp = Float64(0.5 * Float64(x * x));
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.2e-141)
		tmp = 0.5 * (x * x);
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.2e-141], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.20000000000000009e-141

   1. Initial program 11.5%

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

   3. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 5.8%

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

   5. Simplified 5.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 12.1%

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

   8. Simplified 12.1%

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

   if -2.20000000000000009e-141 < x

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos \varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \cos \varepsilon + -1 \]

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

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

      4. cos-lowering-cos.f64 61.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

   5. Simplified 61.3%

      \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} \]
   7. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 63.6%

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

   8. Simplified 63.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

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

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

    13. unpow2 N/A

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

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

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

    15. *-commutative N/A

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

    16. associate-*r* N/A

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

    17. *-commutative N/A

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

    18. associate-*r* N/A

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

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

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

11. Simplified 96.7%

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

12. Taylor expanded in eps around 0

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

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

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. unsub-neg N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 96.6%

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

14. Simplified 96.6%

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

Alternative 14: 78.3% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0 - \varepsilon \cdot x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* eps x)))

C

double code(double x, double eps) {
	return 0.0 - (eps * x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - (eps * x)
end function

Java

public static double code(double x, double eps) {
	return 0.0 - (eps * x);
}

Python

def code(x, eps):
	return 0.0 - (eps * x)

Julia

function code(x, eps)
	return Float64(0.0 - Float64(eps * x))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - (eps * x);
end

Wolfram

code[x_, eps_] := N[(0.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in eps around 0

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

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

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {\varepsilon}^{2}\right)\right)\right) - 1\right)\right)}\right) \]

8. Simplified 97.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(-0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right)\right) + x \cdot \left(x \cdot \left(\varepsilon \cdot \left(0.25 + \varepsilon \cdot \left(\varepsilon \cdot -0.020833333333333332\right)\right) + x \cdot \left(0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot -0.027777777777777776\right)\right)\right) + \left(-1 + \varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right)\right)\right)\right)} \]

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

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

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

    13. unpow2 N/A

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

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

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

    15. *-commutative N/A

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

    16. associate-*r* N/A

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

    17. *-commutative N/A

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

    18. associate-*r* N/A

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

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

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

11. Simplified 96.7%

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

12. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\varepsilon \cdot x\right) \]

    2. neg-sub0 N/A

       \[\leadsto 0 - \color{blue}{\varepsilon \cdot x} \]

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

       \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\varepsilon \cdot x\right)}\right) \]

    4. *-lowering-*.f64 75.0%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\varepsilon, \color{blue}{x}\right)\right) \]

14. Simplified 75.0%

    \[\leadsto \color{blue}{0 - \varepsilon \cdot x} \]
15. Add Preprocessing

Alternative 15: 51.3% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos \varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + -1 \]

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

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

   4. cos-lowering-cos.f64 44.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

5. Simplified 44.4%

   \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} \]
7. Step-by-step derivation

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 45.6%

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

8. Simplified 45.6%

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

9. Final simplification 45.6%

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

Alternative 16: 50.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 46.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos \varepsilon + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + -1 \]

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

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

   4. cos-lowering-cos.f64 44.4%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

5. Simplified 44.4%

   \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

6. Taylor expanded in eps around 0

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

8. Simplified 44.3%

   \[\leadsto \color{blue}{1} + -1 \]
9. Step-by-step derivation

   1. metadata-eval 44.3%

      \[\leadsto 0 \]

10. Applied egg-rr 44.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((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) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :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 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (cos (+ x eps)) (cos x)))