2sin (example 3.3)

Percentage Accurate: 62.2% → 99.8%

Time: 13.0s

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: 62.2% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * -1.5500992063492063e-6 + 0.00026041666666666666), $MachinePrecision]), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $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. *-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. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.00026041666666666666 + -0.020833333333333332), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $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. *-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. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $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. *-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. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 99.8

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

13. Applied rewrites 99.8%

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

14. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * 0.5), $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. *-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. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 99.7

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

13. Applied rewrites 99.7%

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

14. Final simplification 99.7%

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

Alternative 5: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.4%

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

Alternative 6: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[Cos[x], $MachinePrecision]), $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. lower-*.f64 N/A

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

   2. lower-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 7: 98.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, x \cdot 0.08333333333333333, -0.5\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (* x (fma x (* x 0.08333333333333333) -0.5))
   (fma
    (* x x)
    (fma x (* x (fma x (* x -0.001388888888888889) 0.041666666666666664)) -0.5)
    1.0))))

C

double code(double x, double eps) {
	return eps * fma(eps, (x * fma(x, (x * 0.08333333333333333), -0.5)), fma((x * x), fma(x, (x * fma(x, (x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, Float64(x * fma(x, Float64(x * 0.08333333333333333), -0.5)), fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * -0.001388888888888889), 0.041666666666666664)), -0.5), 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(x * N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

12. Taylor expanded in x around 0

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)\right) \]
13. Step-by-step derivation

14. Applied rewrites 98.9%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, -0.5\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right) \]
15. Add Preprocessing

Alternative 8: 98.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

Alternative 9: 98.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.7%

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

Alternative 10: 98.3% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.7%

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

Alternative 11: 98.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.7%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x \cdot -0.5, \color{blue}{\varepsilon + x}, 1\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.6%

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

Alternative 12: 97.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 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 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.1%

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

9. Final simplification 98.1%

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

Alternative 13: 97.9% accurate, 34.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.1%

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

Alternative 14: 5.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

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. 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 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. sin-neg N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\sin x - \sin x} \]

   3. +-inverses 5.3

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 5.3%

   \[\leadsto \color{blue}{0} \]
8. 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 2024226 
(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)))