2cos (problem 3.3.5)

Percentage Accurate: 53.2% → 99.7%

Time: 19.0s

Alternatives: 14

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: 53.2% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. 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. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-+.f64 N/A

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

   5. +-rgt-identity 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in eps around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. associate-*r/ N/A

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

   15. distribute-frac-neg N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-/.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f64 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps * Float64(0.5 + Float64(x / eps)))) * 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[(eps * N[(0.5 + N[(x / eps), $MachinePrecision]), $MachinePrecision]), $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 49.8%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-+.f64 N/A

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

   5. +-rgt-identity 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in eps around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. associate-*r/ N/A

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

   15. distribute-frac-neg N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-/.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. sub-neg N/A

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

    7. *-commutative N/A

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

    8. unpow2 N/A

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

    9. associate-*l* N/A

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

    10. metadata-eval N/A

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

    11. lower-fma.f64 N/A

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

    12. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-+.f64 N/A

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

   5. +-rgt-identity 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in eps around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. associate-*r/ N/A

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

   15. distribute-frac-neg N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-/.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 99.5

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

12. Applied rewrites 99.5%

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

13. Final simplification 99.5%

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

Alternative 6: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-+.f64 N/A

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

   5. +-rgt-identity 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in eps around inf

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. associate-*r/ N/A

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

   15. distribute-frac-neg N/A

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

   16. mul-1-neg N/A

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

   17. remove-double-neg N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-/.f64 99.6

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 99.4

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 7: 80.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{1}{\frac{-1}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (sin x) (/ 1.0 (/ -1.0 eps))))

C

double code(double x, double eps) {
	return sin(x) * (1.0 / (-1.0 / eps));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return Math.sin(x) * (1.0 / (-1.0 / eps));
}

Python

def code(x, eps):
	return math.sin(x) * (1.0 / (-1.0 / eps))

Julia

function code(x, eps)
	return Float64(sin(x) * Float64(1.0 / Float64(-1.0 / eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(x) * (1.0 / (-1.0 / eps));
end

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

7. Applied rewrites 79.8%

   \[\leadsto \sin x \cdot \frac{\varepsilon \cdot \varepsilon}{\color{blue}{-\varepsilon}} \]
8. Step-by-step derivation

9. Applied rewrites 80.2%

   \[\leadsto \sin x \cdot \frac{1}{\color{blue}{\frac{-1}{\varepsilon}}} \]
10. Add Preprocessing

Alternative 8: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Final simplification 80.2%

   \[\leadsto -\varepsilon \cdot \sin x \]
7. Add Preprocessing

Alternative 9: 79.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left(-\varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (- eps)
  (fma
   (fma
    x
    (* x (fma (* x x) -0.0001984126984126984 0.008333333333333333))
    -0.16666666666666666)
   (* x (* x x))
   x)))

C

double code(double x, double eps) {
	return -eps * fma(fma(x, (x * fma((x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), (x * (x * x)), x);
}

Julia

function code(x, eps)
	return Float64(Float64(-eps) * fma(fma(x, Float64(x * fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), Float64(x * Float64(x * x)), x))
end

Wolfram

code[x_, eps_] := N[((-eps) * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\varepsilon}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 79.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \cdot \left(-\color{blue}{\varepsilon}\right) \]

9. Final simplification 79.7%

   \[\leadsto \left(-\varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \]
10. Add Preprocessing

Alternative 10: 79.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(-0.008333333333333333, x \cdot x, 0.16666666666666666\right), -\varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (* eps (fma -0.008333333333333333 (* x x) 0.16666666666666666))
   (- eps))))

C

double code(double x, double eps) {
	return x * fma((x * x), (eps * fma(-0.008333333333333333, (x * x), 0.16666666666666666)), -eps);
}

Julia

function code(x, eps)
	return Float64(x * fma(Float64(x * x), Float64(eps * fma(-0.008333333333333333, Float64(x * x), 0.16666666666666666)), Float64(-eps)))
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(eps * N[(-0.008333333333333333 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{120} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{6} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 79.6%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \varepsilon \cdot \mathsf{fma}\left(-0.008333333333333333, x \cdot x, 0.16666666666666666\right), -\varepsilon\right)} \]
9. Add Preprocessing

Alternative 11: 79.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.6%

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

9. Final simplification 79.6%

   \[\leadsto \left(-\varepsilon\right) \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot -0.16666666666666666\right), x\right) \]
10. Add Preprocessing

Alternative 12: 79.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.6%

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

Alternative 13: 79.3% 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 49.8%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 80.2

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.3%

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

Alternative 14: 51.9% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ -1 + 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.8%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 48.7

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

5. Applied rewrites 48.7%

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

6. Taylor expanded in eps around 0

   \[\leadsto 1 + -1 \]
7. Step-by-step derivation

8. Applied rewrites 48.6%

   \[\leadsto 1 + -1 \]

9. Final simplification 48.6%

   \[\leadsto -1 + 1 \]
10. 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 2024226 
(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)))