2sin (example 3.3)

Percentage Accurate: 62.6% → 99.7%

Time: 14.8s

Alternatives: 13

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: 62.6% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathsf{fma}\left(t\_0, -\sin x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.0026041666666666665, -0.125\right), 1\right) \cdot \cos x\right) \cdot \left(t\_0 \cdot 2\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (*
    (fma
     t_0
     (- (sin x))
     (*
      (fma (* eps eps) (fma eps (* eps 0.0026041666666666665) -0.125) 1.0)
      (cos x)))
    (* t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return fma(t_0, -sin(x), (fma((eps * eps), fma(eps, (eps * 0.0026041666666666665), -0.125), 1.0) * cos(x))) * (t_0 * 2.0);
}

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(fma(t_0, Float64(-sin(x)), Float64(fma(Float64(eps * eps), fma(eps, Float64(eps * 0.0026041666666666665), -0.125), 1.0) * cos(x))) * Float64(t_0 * 2.0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * (-N[Sin[x], $MachinePrecision]) + N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * 0.0026041666666666665), $MachinePrecision] + -0.125), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.1%

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

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \cos \left(\mathsf{fma}\left(x, 2, \varepsilon\right) \cdot 0.5\right)\right) \cdot 2} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. +-rgt-identity N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. cos-sum N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-rgt-identity N/A

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

   18. lift-sin.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}\right)\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}\right) + 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   2. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   3. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   5. sub-neg N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{384} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   7. metadata-eval N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   9. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   10. lower-*.f64 100.0

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.0026041666666666665, -0.125\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot 2 \]

9. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot 2 \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), -\sin x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.0026041666666666665, -0.125\right), 1\right) \cdot \cos x\right) \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot 2\right)} \]
12. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos \left(0.5 \cdot \left(x \cdot 2\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (sin (* 0.5 eps))
   (-
    (*
     (cos (* 0.5 (* x 2.0)))
     (fma (* eps eps) (fma (* eps eps) 0.0026041666666666665 -0.125) 1.0))
    (* eps (* (sin x) (fma eps (* eps -0.020833333333333332) 0.5)))))))

C

double code(double x, double eps) {
	return 2.0 * (sin((0.5 * eps)) * ((cos((0.5 * (x * 2.0))) * fma((eps * eps), fma((eps * eps), 0.0026041666666666665, -0.125), 1.0)) - (eps * (sin(x) * fma(eps, (eps * -0.020833333333333332), 0.5)))));
}

Julia

function code(x, eps)
	return Float64(2.0 * Float64(sin(Float64(0.5 * eps)) * Float64(Float64(cos(Float64(0.5 * Float64(x * 2.0))) * fma(Float64(eps * eps), fma(Float64(eps * eps), 0.0026041666666666665, -0.125), 1.0)) - Float64(eps * Float64(sin(x) * fma(eps, Float64(eps * -0.020833333333333332), 0.5))))))
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.0026041666666666665 + -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

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

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \cos \left(\mathsf{fma}\left(x, 2, \varepsilon\right) \cdot 0.5\right)\right) \cdot 2} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. +-rgt-identity N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. cos-sum N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-rgt-identity N/A

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

   18. lift-sin.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}\right)\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}\right) + 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   2. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   3. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384} \cdot {\varepsilon}^{2} - \frac{1}{8}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   5. sub-neg N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{384} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{384}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   7. metadata-eval N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \frac{1}{384} + \color{blue}{\frac{-1}{8}}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{384}, \frac{-1}{8}\right)}, 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   9. unpow2 N/A

      \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot 2 \]

   10. lower-*.f64 99.7

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.0026041666666666665, -0.125\right), 1\right) - \sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot 2 \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right)} - \sin \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right) \cdot 2 \]

10. Taylor expanded in eps around 0

    \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \color{blue}{\varepsilon \cdot \left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \frac{1}{2} \cdot \sin x\right)}\right)\right) \cdot 2 \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \sin x\right) \cdot \frac{-1}{48}} + \frac{1}{2} \cdot \sin x\right)\right)\right) \cdot 2 \]

    2. associate-*r* N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\sin x \cdot \frac{-1}{48}\right)} + \frac{1}{2} \cdot \sin x\right)\right)\right) \cdot 2 \]

    3. *-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(\frac{-1}{48} \cdot \sin x\right)} + \frac{1}{2} \cdot \sin x\right)\right)\right) \cdot 2 \]

    4. +-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \sin x\right)\right)}\right)\right) \cdot 2 \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \color{blue}{\varepsilon \cdot \left(\frac{1}{2} \cdot \sin x + {\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \sin x\right)\right)}\right)\right) \cdot 2 \]

    6. associate-*r* N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\frac{1}{2} \cdot \sin x + \color{blue}{\left({\varepsilon}^{2} \cdot \frac{-1}{48}\right) \cdot \sin x}\right)\right)\right) \cdot 2 \]

    7. *-commutative N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\frac{1}{2} \cdot \sin x + \color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2}\right)} \cdot \sin x\right)\right)\right) \cdot 2 \]

    8. distribute-rgt-out N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \color{blue}{\left(\sin x \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \cdot 2 \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \color{blue}{\left(\sin x \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \cdot 2 \]

    10. lower-sin.f64 N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\color{blue}{\sin x} \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot 2 \]

    11. +-commutative N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right)}\right)\right)\right) \cdot 2 \]

    12. *-commutative N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{48}} + \frac{1}{2}\right)\right)\right)\right) \cdot 2 \]

    13. unpow2 N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{-1}{48} + \frac{1}{2}\right)\right)\right)\right) \cdot 2 \]

    14. associate-*l* N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{-1}{48}\right)} + \frac{1}{2}\right)\right)\right)\right) \cdot 2 \]

    15. *-commutative N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{-1}{48} \cdot \varepsilon\right)} + \frac{1}{2}\right)\right)\right)\right) \cdot 2 \]

    16. lower-fma.f64 N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{48} \cdot \varepsilon, \frac{1}{2}\right)}\right)\right)\right) \cdot 2 \]

    17. *-commutative N/A

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot \frac{1}{2}\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{384}, \frac{-1}{8}\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{-1}{48}}, \frac{1}{2}\right)\right)\right)\right) \cdot 2 \]

    18. lower-*.f64 99.7

        \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot -0.020833333333333332}, 0.5\right)\right)\right)\right) \cdot 2 \]

12. Applied rewrites 99.7%

    \[\leadsto \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \left(\cos \left(\left(x \cdot 2\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right) - \color{blue}{\varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right)}\right)\right) \cdot 2 \]

13. Final simplification 99.7%

    \[\leadsto 2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos \left(0.5 \cdot \left(x \cdot 2\right)\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.0026041666666666665, -0.125\right), 1\right) - \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right)\right)\right) \]
14. 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 2024223 
(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)))