2cos (problem 3.3.5)

Percentage Accurate: 52.0% → 99.6%

Time: 18.4s

Alternatives: 15

Speedup: 25.9×

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: 52.0% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + x\right)\\ \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin t\_0, \cos \left(\varepsilon \cdot 0.5\right), \cos t\_0 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right)\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x x))))
   (*
    (*
     (sin (* eps 0.5))
     (fma
      (sin t_0)
      (cos (* eps 0.5))
      (*
       (cos t_0)
       (*
        eps
        (fma
         (* eps eps)
         (fma eps (* eps 0.00026041666666666666) -0.020833333333333332)
         0.5)))))
    -2.0)))

C

double code(double x, double eps) {
	double t_0 = 0.5 * (x + x);
	return (sin((eps * 0.5)) * fma(sin(t_0), cos((eps * 0.5)), (cos(t_0) * (eps * fma((eps * eps), fma(eps, (eps * 0.00026041666666666666), -0.020833333333333332), 0.5))))) * -2.0;
}

Julia

function code(x, eps)
	t_0 = Float64(0.5 * Float64(x + x))
	return Float64(Float64(sin(Float64(eps * 0.5)) * fma(sin(t_0), cos(Float64(eps * 0.5)), Float64(cos(t_0) * Float64(eps * fma(Float64(eps * eps), fma(eps, Float64(eps * 0.00026041666666666666), -0.020833333333333332), 0.5))))) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$0], $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * 0.00026041666666666666), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 53.5%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]
5. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\left(x + \left(x + \varepsilon\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \cdot -2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)}\right)\right) \cdot -2 \]

   5. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right)\right) \cdot -2 \]

   6. associate-+r+ N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \cdot -2 \]

   7. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\left(x + x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   8. +-rgt-identity N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right)\right) \cdot -2 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right)\right) \cdot -2 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right) \cdot \frac{1}{2}}\right)\right) \cdot -2 \]

   11. sin-sum N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) + \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot -2 \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot -2 \]

   13. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\left(x + x\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   15. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{1}{2}\right), \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   16. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   17. lift-+.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   18. +-rgt-identity N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\color{blue}{\varepsilon} \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   19. lift-sin.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}\right)\right) \cdot -2 \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}\right)\right) \cdot -2 \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(\varepsilon \cdot 0.5\right), \cos \left(\left(x + x\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right) \cdot -2 \]

7. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}\right)\right) \cdot -2 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}\right)\right) \cdot -2 \]

   2. +-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right)}\right)\right)\right) \cdot -2 \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)}\right)\right)\right) \cdot -2 \]

   4. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   6. sub-neg N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   7. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{3840}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   8. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{3840} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   9. associate-*l* N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   10. metadata-eval N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{3840}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{3840}, \frac{-1}{48}\right)}, \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

   12. lower-*.f64 99.8

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(\varepsilon \cdot 0.5\right), \cos \left(\left(x + x\right) \cdot 0.5\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.00026041666666666666}, -0.020833333333333332\right), 0.5\right)\right)\right)\right) \cdot -2 \]

9. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(\varepsilon \cdot 0.5\right), \cos \left(\left(x + x\right) \cdot 0.5\right) \cdot \color{blue}{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)}\right)\right) \cdot -2 \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

       \[\leadsto \left(\sin \left(\color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{3840}, \frac{-1}{48}\right), \frac{1}{2}\right)\right)\right)\right) \cdot -2 \]

    2. +-rgt-identity 99.8

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \left(x + x\right)\right), \cos \left(\varepsilon \cdot 0.5\right), \cos \left(0.5 \cdot \left(x + x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right)\right) \cdot -2 \]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right) \cdot -2} \]
5. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(x + \left(x + \varepsilon\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\left(x + \left(x + \varepsilon\right)\right) \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(x + \left(x + \varepsilon\right)\right)\right)}\right) \cdot -2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)}\right)\right) \cdot -2 \]

   5. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right)\right) \cdot -2 \]

   6. associate-+r+ N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right)\right) \cdot -2 \]

   7. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\left(x + x\right) \cdot \frac{1}{2} + \varepsilon \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   8. +-rgt-identity N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right)\right) \cdot -2 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right)\right) \cdot -2 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(x + x\right) \cdot \frac{1}{2} + \color{blue}{\left(\varepsilon + 0\right) \cdot \frac{1}{2}}\right)\right) \cdot -2 \]

   11. sin-sum N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) + \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot -2 \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot -2 \]

   13. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\left(x + x\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   15. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{1}{2}\right), \cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   16. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}, \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   17. lift-+.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\color{blue}{\left(\varepsilon + 0\right)} \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   18. +-rgt-identity N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\color{blue}{\varepsilon} \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)\right)\right) \cdot -2 \]

   19. lift-sin.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}\right)\right) \cdot -2 \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right)}\right)\right) \cdot -2 \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(\varepsilon \cdot 0.5\right), \cos \left(\left(x + x\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right) \cdot -2 \]

7. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \color{blue}{\varepsilon \cdot \left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{2} \cdot \cos x\right)}\right)\right) \cdot -2 \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \cos x\right) \cdot \frac{-1}{48}} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   2. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \cos x\right) \cdot \frac{-1}{48} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   3. associate-*l* N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right)} \cdot \frac{-1}{48} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   4. associate-*r* N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{48}\right)} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   5. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{48} \cdot \left(\varepsilon \cdot \cos x\right)\right)} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{48} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \frac{1}{2} \cdot \cos x\right)}\right)\right) \cdot -2 \]

   7. associate-*r* N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \cos x\right)} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   8. associate-*r* N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{48} \cdot \varepsilon\right)\right) \cdot \cos x} + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   9. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{-1}{48}\right)}\right) \cdot \cos x + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   10. associate-*r* N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{48}\right)} \cdot \cos x + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   11. unpow2 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\left(\color{blue}{{\varepsilon}^{2}} \cdot \frac{-1}{48}\right) \cdot \cos x + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   12. *-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2}\right)} \cdot \cos x + \frac{1}{2} \cdot \cos x\right)\right)\right) \cdot -2 \]

   13. distribute-rgt-out N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \color{blue}{\left(\cos x \cdot \left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right)\right)}\right)\right) \cdot -2 \]

   14. +-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\cos x \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)}\right)\right)\right) \cdot -2 \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \color{blue}{\left(\cos x \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \cdot -2 \]

   16. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\color{blue}{\cos x} \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot -2 \]

   17. +-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot \frac{1}{2}\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right), \varepsilon \cdot \left(\cos x \cdot \color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right)}\right)\right)\right) \cdot -2 \]

9. Applied rewrites 99.5%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin \left(\left(x + x\right) \cdot 0.5\right), \cos \left(\varepsilon \cdot 0.5\right), \color{blue}{\varepsilon \cdot \left(\cos x \cdot \mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right)\right)}\right)\right) \cdot -2 \]

10. Final simplification 99.5%

    \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \mathsf{fma}\left(\sin \left(0.5 \cdot \left(x + x\right)\right), \cos \left(\varepsilon \cdot 0.5\right), \varepsilon \cdot \left(\cos x \cdot \mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right)\right)\right)\right) \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× 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]

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024223 
(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))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

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