2cos (problem 3.3.5)

Percentage Accurate: 52.9% → 99.7%

Time: 18.1s

Alternatives: 12

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: 52.9% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right)} \cdot -2 \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

      \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot -2 \]

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{645120}, \frac{1}{3840}\right)}, \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot -2 \]

   14. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{645120}, \frac{1}{3840}\right), \frac{-1}{48}\right), \frac{1}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right) \cdot -2 \]

   15. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

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

Alternative 5: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.9

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

10. Applied rewrites 98.9%

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

11. Taylor expanded in eps around inf

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

    1. associate-*r* N/A

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

    2. metadata-eval N/A

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

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

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower-sin.f64 N/A

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. mul-1-neg N/A

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

    12. lower-neg.f64 98.9

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

13. Applied rewrites 98.9%

    \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(-\varepsilon\right)} \]
14. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.6

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

8. Applied rewrites 98.6%

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

Alternative 7: 98.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-neg-in N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

   10. distribute-rgt-out N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

8. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.5 - x, x \cdot \left(x \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 8: 98.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4}, \frac{1}{6} \cdot x\right)}, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   9. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   11. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, x \cdot \frac{1}{6}\right), -1\right), \color{blue}{\varepsilon \cdot \frac{-1}{2}}\right) \]

   12. lower-*.f64 97.5

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

8. Applied rewrites 97.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -1\right), \varepsilon \cdot -0.5\right)} \]
9. Add Preprocessing

Alternative 9: 97.8% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{4} \cdot {\varepsilon}^{2}}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-1}{2}\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{-1}{2}\right)}\right) \]

   18. lower-*.f64 97.2

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \left(\varepsilon \cdot \varepsilon\right), -\varepsilon\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5\right)}\right) \]

8. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \left(\varepsilon \cdot \varepsilon\right), -\varepsilon\right), \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)} \]

9. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 97.3

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

11. Applied rewrites 97.3%

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

Alternative 10: 97.6% accurate, 14.8× 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 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.2

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

8. Applied rewrites 97.2%

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

Alternative 11: 78.8% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{4} \cdot {\varepsilon}^{2}}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-1}{2}\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{2}\right)}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4} \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{neg}\left(\varepsilon\right)\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{-1}{2}\right)}\right) \]

   18. lower-*.f64 97.2

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \left(\varepsilon \cdot \varepsilon\right), -\varepsilon\right), \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5\right)}\right) \]

8. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25 \cdot \left(\varepsilon \cdot \varepsilon\right), -\varepsilon\right), \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. mul-1-neg N/A

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

    2. distribute-rgt-neg-in N/A

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

    3. mul-1-neg N/A

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

    4. lower-*.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 78.7

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

11. Applied rewrites 78.7%

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

Alternative 12: 51.4% accurate, 207.0× 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 54.8%

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

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

   2. metadata-eval N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cos.f64 53.1

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

5. Applied rewrites 53.1%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 53.0%

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

   1. metadata-eval 53.0

      \[\leadsto \color{blue}{0} \]

10. Applied rewrites 53.0%

    \[\leadsto \color{blue}{0} \]
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: 98.8% 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 2024219 
(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)))