2sin (example 3.3)

Percentage Accurate: 62.4% → 99.8%

Time: 15.1s

Alternatives: 14

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: 62.4% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin \varepsilon, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), \sin x \cdot -0.5\right), 0\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin eps)
  (cos x)
  (fma
   (* eps eps)
   (fma
    (* eps eps)
    (* (sin x) (fma eps (* eps -0.001388888888888889) 0.041666666666666664))
    (* (sin x) -0.5))
   0.0)))

C

double code(double x, double eps) {
	return fma(sin(eps), cos(x), fma((eps * eps), fma((eps * eps), (sin(x) * fma(eps, (eps * -0.001388888888888889), 0.041666666666666664)), (sin(x) * -0.5)), 0.0));
}

Julia

function code(x, eps)
	return fma(sin(eps), cos(x), fma(Float64(eps * eps), fma(Float64(eps * eps), Float64(sin(x) * fma(eps, Float64(eps * -0.001388888888888889), 0.041666666666666664)), Float64(sin(x) * -0.5)), 0.0))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. sin-sum N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. sin-neg N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-neg.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\sin x + \sin \left(\mathsf{neg}\left(x\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \left(\sin x + \sin \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

   3. sin-neg N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \left(\sin x + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

   4. unsub-neg N/A

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

   5. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \color{blue}{0}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right), 0\right)}\right) \]

7. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5 \cdot \sin x\right), 0\right)}\right) \]

8. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), \sin x \cdot -0.5\right), 0\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), \sin x \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (fma
   (* eps eps)
   (* eps (fma (* eps eps) 0.008333333333333333 -0.16666666666666666))
   eps)
  (*
   (* eps eps)
   (fma
    (* eps eps)
    (* (sin x) (fma eps (* eps -0.001388888888888889) 0.041666666666666664))
    (* (sin x) -0.5)))))

C

double code(double x, double eps) {
	return fma(cos(x), fma((eps * eps), (eps * fma((eps * eps), 0.008333333333333333, -0.16666666666666666)), eps), ((eps * eps) * fma((eps * eps), (sin(x) * fma(eps, (eps * -0.001388888888888889), 0.041666666666666664)), (sin(x) * -0.5))));
}

Julia

function code(x, eps)
	return fma(cos(x), fma(Float64(eps * eps), Float64(eps * fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666)), eps), Float64(Float64(eps * eps) * fma(Float64(eps * eps), Float64(sin(x) * fma(eps, Float64(eps * -0.001388888888888889), 0.041666666666666664)), Float64(sin(x) * -0.5))))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. sin-sum N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. sin-neg N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\left(\sin x + \sin \left(\mathsf{neg}\left(x\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \left(\sin x + \sin \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

   3. sin-neg N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \left(\sin x + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

   4. unsub-neg N/A

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

   5. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, {\varepsilon}^{2} \cdot \left(\frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right)\right) + \color{blue}{0}\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{720} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{24} \cdot \sin x\right), 0\right)}\right) \]

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5 \cdot \sin x\right), 0\right)}\right) \]

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \varepsilon \cdot 1}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\varepsilon}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right)}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}, \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \left({\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right)\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

   14. lower-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right), \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5 \cdot \sin x\right), 0\right)\right) \]

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right)\right), \varepsilon\right)}, \cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5 \cdot \sin x\right), 0\right)\right) \]
11. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right), \color{blue}{\cos x}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right) \]

    2. lift-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right) \cdot \cos x + \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\cos x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right)} + \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{120}, \frac{-1}{6}\right)\right), \varepsilon\right), \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2} \cdot \sin x\right), 0\right)\right)} \]

12. Applied rewrites 99.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.001388888888888889, 0.041666666666666664\right), \sin x \cdot -0.5\right)\right)} \]
13. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2024228 
(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 (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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

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

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