2sin (example 3.3)

Percentage Accurate: 62.5% → 99.9%

Time: 11.5s

Alternatives: 11

Speedup: 34.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.8%

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

Alternative 4: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.8%

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

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. +-inverses N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. neg-mul-1 N/A

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

   5. metadata-eval N/A

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

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

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

   7. distribute-neg-in N/A

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

   8. +-commutative N/A

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

   9. cos-neg N/A

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

   10. *-rgt-identity N/A

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

   11. *-lft-identity N/A

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

   12. *-inverses N/A

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

   13. /-rgt-identity N/A

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

   14. times-frac N/A

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

   15. *-rgt-identity N/A

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

   16. associate-*r/ N/A

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

   17. associate-*r/ N/A

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

   18. distribute-lft-in N/A

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

   19. lower-cos.f64 N/A

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

   20. +-commutative N/A

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

   21. distribute-lft-in N/A

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 6: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   3. lower-cos.f64 99.2

      \[\leadsto \color{blue}{\cos x} \cdot \varepsilon \]

5. Applied rewrites 99.2%

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

Alternative 7: 98.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x) -0.5)
   (* x x)
   1.0)
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5), Float64(x * x), 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   3. lower-cos.f64 99.2

      \[\leadsto \color{blue}{\cos x} \cdot \varepsilon \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 8: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0) eps))

C

double code(double x, double eps) {
	return fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0) * eps)
end

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   3. lower-cos.f64 99.2

      \[\leadsto \color{blue}{\cos x} \cdot \varepsilon \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 9: 98.2% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

Alternative 10: 98.2% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   3. lower-cos.f64 99.2

      \[\leadsto \color{blue}{\cos x} \cdot \varepsilon \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

Alternative 11: 97.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 1.0 eps))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0 * eps

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

   3. lower-cos.f64 99.2

      \[\leadsto \color{blue}{\cos x} \cdot \varepsilon \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

   \[\leadsto 1 \cdot \varepsilon \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

  (- (sin (+ x eps)) (sin x)))