2sin (example 3.3)

Percentage Accurate: 63.0% → 99.8%

Time: 11.9s

Alternatives: 8

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: 63.0% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.48015873015873 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, -0.001388888888888889\right), \varepsilon \cdot \varepsilon, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (*
   (*
    (*
     (fma
      (fma
       (fma 2.48015873015873e-5 (* eps eps) -0.001388888888888889)
       (* eps eps)
       0.041666666666666664)
      (* eps eps)
      -0.5)
     eps)
    eps)
   (sin x))))

C

double code(double x, double eps) {
	return fma(sin(eps), cos(x), (((fma(fma(fma(2.48015873015873e-5, (eps * eps), -0.001388888888888889), (eps * eps), 0.041666666666666664), (eps * eps), -0.5) * eps) * eps) * sin(x)));
}

Julia

function code(x, eps)
	return fma(sin(eps), cos(x), Float64(Float64(Float64(fma(fma(fma(2.48015873015873e-5, Float64(eps * eps), -0.001388888888888889), Float64(eps * eps), 0.041666666666666664), Float64(eps * eps), -0.5) * eps) * eps) * sin(x)))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(N[(N[(N[(2.48015873015873e-5 * N[(eps * eps), $MachinePrecision] + -0.001388888888888889), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

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

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

   4. sin-sum N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around inf

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

   1. sub-neg N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + {\varepsilon}^{2} \cdot \left(\frac{1}{40320} \cdot {\varepsilon}^{2} - \frac{1}{720}\right)\right) - \frac{1}{2}\right)}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.48015873015873 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, -0.001388888888888889\right), \varepsilon \cdot \varepsilon, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right) \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.48015873015873 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, -0.001388888888888889\right), \varepsilon \cdot \varepsilon, 0.041666666666666664\right), \varepsilon \cdot \varepsilon, -0.5\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \sin x\right) \]
12. Add Preprocessing

Alternative 2: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

Julia

function code(x, eps)
	return Float64(cos(fma(eps, -0.5, Float64(-x))) * Float64(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[Cos[N[(eps * -0.5 + (-x)), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[(N[(N[(0.00026041666666666666 * N[(eps * eps), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-cos.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 99.9

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

13. Applied rewrites 99.9%

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

14. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in eps around inf

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*r* N/A

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

    4. metadata-eval N/A

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

    5. neg-mul-1 N/A

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

    6. metadata-eval N/A

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

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

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

    8. distribute-neg-in N/A

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

    9. cos-neg N/A

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

    10. lower-cos.f64 N/A

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

    11. +-commutative N/A

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

    12. lower-fma.f64 99.8

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

13. Applied rewrites 99.8%

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

14. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. lower-cos.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in eps around 0

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

    1. lower-*.f64 99.7

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

13. Applied rewrites 99.7%

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

Alternative 5: 98.9% 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 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.4

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

5. Applied rewrites 99.4%

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

Alternative 6: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.0

      \[\leadsto \color{blue}{\sin \varepsilon} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sin \varepsilon} \]
6. Add Preprocessing

Alternative 7: 97.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \left(\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.020833333333333332, 0.5\right) \cdot \varepsilon\right) \cdot 2\right) \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return 1.0 * ((fma((eps * eps), -0.020833333333333332, 0.5) * eps) * 2.0);
}

Julia

function code(x, eps)
	return Float64(1.0 * Float64(Float64(fma(Float64(eps * eps), -0.020833333333333332, 0.5) * eps) * 2.0))
end

Wolfram

code[x_, eps_] := N[(1.0 * N[(N[(N[(N[(eps * eps), $MachinePrecision] * -0.020833333333333332 + 0.5), $MachinePrecision] * eps), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.9%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. cos-neg N/A

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

   3. lower-cos.f64 99.4

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 99.0%

    \[\leadsto \left(\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.020833333333333332, 0.5\right) \cdot \varepsilon\right) \cdot 2\right) \cdot 1 \]

14. Final simplification 99.0%

    \[\leadsto 1 \cdot \left(\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.020833333333333332, 0.5\right) \cdot \varepsilon\right) \cdot 2\right) \]
15. Add Preprocessing

Alternative 8: 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 62.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

   \[\leadsto 1 \cdot \varepsilon \]
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 2024270 
(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)))