2cos (problem 3.3.5)

Percentage Accurate: 52.1% → 99.5%

Time: 18.3s

Alternatives: 15

Speedup: 25.9×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right), \varepsilon, \left(-0.5 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* (sin x) (fma eps (* eps 0.16666666666666666) -1.0))
  eps
  (* (* -0.5 (cos x)) (* eps eps))))

C

double code(double x, double eps) {
	return fma((sin(x) * fma(eps, (eps * 0.16666666666666666), -1.0)), eps, ((-0.5 * cos(x)) * (eps * eps)));
}

Julia

function code(x, eps)
	return fma(Float64(sin(x) * fma(eps, Float64(eps * 0.16666666666666666), -1.0)), eps, Float64(Float64(-0.5 * cos(x)) * Float64(eps * eps)))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. *-lowering-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right), \varepsilon, \left(-0.5 \cdot \cos x\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right), \varepsilon, \left(-0.5 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. accelerator-lowering-fma.f64 99.8

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

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 99.8

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

10. Simplified 99.8%

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

Alternative 3: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(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[Sin[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 52.9%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. accelerator-lowering-fma.f64 99.8

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

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 99.7

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. accelerator-lowering-fma.f64 99.8

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

7. Simplified 99.8%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.5

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma x (* x 0.25) -0.5)
   (*
    (fma eps (* eps 0.16666666666666666) -1.0)
    (fma
     (fma
      (* x x)
      (fma (* x x) -0.0001984126984126984 0.008333333333333333)
      -0.16666666666666666)
     (* x (* x x))
     x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(x, (x * 0.25), -0.5), (fma(eps, (eps * 0.16666666666666666), -1.0) * fma(fma((x * x), fma((x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(x, Float64(x * 0.25), -0.5), Float64(fma(eps, Float64(eps * 0.16666666666666666), -1.0) * fma(fma(Float64(x * x), fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   7. *-lowering-*.f64 99.2

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.25}, -0.5\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

8. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, -0.5\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    3. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    7. unpow3 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{x}^{3}} + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, {x}^{3}, x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

11. Simplified 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

12. Final simplification 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \]
13. Add Preprocessing

Alternative 6: 98.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma x (* x 0.25) -0.5)
   (*
    (fma eps (* eps 0.16666666666666666) -1.0)
    (fma
     (fma x (* x 0.008333333333333333) -0.16666666666666666)
     (* x (* x x))
     x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(x, (x * 0.25), -0.5), (fma(eps, (eps * 0.16666666666666666), -1.0) * fma(fma(x, (x * 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(x, Float64(x * 0.25), -0.5), Float64(fma(eps, Float64(eps * 0.16666666666666666), -1.0) * fma(fma(x, Float64(x * 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   7. *-lowering-*.f64 99.2

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.25}, -0.5\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

8. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, -0.5\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + 1 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    6. unpow3 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    7. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{3} + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{3}, x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    9. sub-neg N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    12. associate-*l* N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \frac{1}{120}\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(x \cdot \left(x \cdot \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    14. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right)}, {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{-1}{6}\right), {x}^{3}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    16. cube-mult N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right), x \cdot \color{blue}{{x}^{2}}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    18. *-lowering-*.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    19. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{-1}{6}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    20. *-lowering-*.f64 99.0

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

11. Simplified 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

12. Final simplification 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\right) \]
13. Add Preprocessing

Alternative 7: 98.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), -\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma x (* x 0.25) -0.5)
   (-
    (fma
     x
     (*
      (* x x)
      (fma
       x
       (* x (fma (* x x) -0.0001984126984126984 0.008333333333333333))
       -0.16666666666666666))
     x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(x, (x * 0.25), -0.5), -fma(x, ((x * x) * fma(x, (x * fma((x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), x));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(x, Float64(x * 0.25), -0.5), Float64(-fma(x, Float64(Float64(x * x) * fma(x, Float64(x * fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision] + (-N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \frac{1}{4}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

   7. *-lowering-*.f64 99.2

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.25}, -0.5\right), \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

8. Simplified 99.2%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, x \cdot 0.25, -0.5\right)}, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + 1 \cdot x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    3. *-lft-identity N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot {x}^{2}\right)} \cdot x + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    5. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left({x}^{2} \cdot x\right)} + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    7. unpow3 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{x}^{3}} + x\right) \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, {x}^{3}, x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right)\right) \]

11. Simplified 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right) \]

12. Taylor expanded in eps around 0

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{-1 \cdot \left(x + {x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\mathsf{neg}\left(\left(x + {x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \]

    2. neg-lowering-neg.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{\mathsf{neg}\left(\left(x + {x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \]

    3. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{neg}\left(\color{blue}{\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x\right)}\right)\right) \]

    4. cube-mult N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{neg}\left(\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{neg}\left(\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{neg}\left(\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x\right)\right)\right) \]

    7. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right), x\right)}\right)\right) \]

14. Simplified 99.0%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), \color{blue}{-\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x\right)}\right) \]
15. Add Preprocessing

Alternative 8: 98.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   -0.5
   (*
    x
    (fma
     x
     (fma
      x
      (* (fma eps (* eps 0.16666666666666666) -1.0) -0.16666666666666666)
      (* eps 0.25))
     -1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, -0.5, (x * fma(x, fma(x, (fma(eps, (eps * 0.16666666666666666), -1.0) * -0.16666666666666666), (eps * 0.25)), -1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, -0.5, Float64(x * fma(x, fma(x, Float64(fma(eps, Float64(eps * 0.16666666666666666), -1.0) * -0.16666666666666666), Float64(eps * 0.25)), -1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * -0.5 + N[(x * N[(x * N[(x * N[(N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(eps * 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \frac{1}{6} \cdot {\varepsilon}^{2}\right)} - 1\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon, \frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)}\right) \]

8. Simplified 98.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)\right)} \]

9. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \frac{1}{6}, -1\right) \cdot \frac{-1}{6}, \varepsilon \cdot \frac{1}{4}\right), \color{blue}{-1}\right)\right) \]
10. Step-by-step derivation

11. Simplified 98.8%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), \color{blue}{-1}\right)\right) \]
12. Add Preprocessing

Alternative 9: 98.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), x \cdot \mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma x (* x 0.25) -0.5)
   (* x (fma 0.16666666666666666 (* x x) -1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(x, (x * 0.25), -0.5), (x * fma(0.16666666666666666, (x * x), -1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(x, Float64(x * 0.25), -0.5), Float64(x * fma(0.16666666666666666, Float64(x * x), -1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(x * N[(x * 0.25), $MachinePrecision] + -0.5), $MachinePrecision] + N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \frac{1}{6} \cdot {\varepsilon}^{2}\right)} - 1\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon, \frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)}\right) \]

8. Simplified 98.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)\right)} \]

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot {x}^{2} - \frac{1}{2}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)} \]

    3. sub-neg N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \frac{1}{4}} + \frac{-1}{2}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} + \frac{-1}{2}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \frac{1}{4}\right)} + \frac{-1}{2}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(\frac{1}{4} \cdot x\right)} + \frac{-1}{2}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    9. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} \cdot x, \frac{-1}{2}\right)}, x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{4}}, \frac{-1}{2}\right), x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{4}}, \frac{-1}{2}\right), x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), \color{blue}{x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)}\right) \]

    13. sub-neg N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), x \cdot \color{blue}{\left(\frac{1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    14. metadata-eval N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), x \cdot \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{-1}\right)\right) \]

    15. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, -1\right)}\right) \]

    16. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{1}{4}, \frac{-1}{2}\right), x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{x \cdot x}, -1\right)\right) \]

    17. *-lowering-*.f64 98.8

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot x}, -1\right)\right) \]

11. Simplified 98.8%

    \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 0.25, -0.5\right), x \cdot \mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right)} \]
12. Add Preprocessing

Alternative 10: 98.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma eps -0.5 (* x (fma x (fma eps 0.25 (* x 0.16666666666666666)) -1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, -0.5, (x * fma(x, fma(eps, 0.25, (x * 0.16666666666666666)), -1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, -0.5, Float64(x * fma(x, fma(eps, 0.25, Float64(x * 0.16666666666666666)), -1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * -0.5 + N[(x * N[(x * N[(eps * 0.25 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \frac{1}{6} \cdot {\varepsilon}^{2}\right)} - 1\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon, \frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)}\right) \]

8. Simplified 98.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)\right)} \]

9. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right) - 1\right)}\right) \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    2. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\left(\frac{1}{6} \cdot x\right) \cdot x + \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right) \cdot x}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    5. distribute-rgt-out N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \color{blue}{-1}\right)\right) \]

    7. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon, -1\right)}\right) \]

    8. +-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{4} \cdot \varepsilon + \frac{1}{6} \cdot x}, -1\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{4}} + \frac{1}{6} \cdot x, -1\right)\right) \]

    10. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4}, \frac{1}{6} \cdot x\right)}, -1\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), -1\right)\right) \]

    12. *-lowering-*.f64 98.8

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, \color{blue}{x \cdot 0.16666666666666666}\right), -1\right)\right) \]

11. Simplified 98.8%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -1\right)}\right) \]
12. Add Preprocessing

Alternative 11: 98.0% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma eps -0.5 (* x (fma 0.16666666666666666 (* x x) -1.0)))))

C

double code(double x, double eps) {
	return eps * fma(eps, -0.5, (x * fma(0.16666666666666666, (x * x), -1.0)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, -0.5, Float64(x * fma(0.16666666666666666, Float64(x * x), -1.0))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * -0.5 + N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, \color{blue}{x \cdot \left(\left(\frac{1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right)\right) - 1\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \frac{1}{6} \cdot {\varepsilon}^{2}\right)} - 1\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon\right) + \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) + \frac{1}{4} \cdot \varepsilon, \frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)}\right) \]

8. Simplified 98.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right) \cdot -0.16666666666666666, \varepsilon \cdot 0.25\right), \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

    3. accelerator-lowering-fma.f64 N/A

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

    4. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{x \cdot x}, -1\right)\right) \]

    5. *-lowering-*.f64 98.7

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot x}, -1\right)\right) \]

11. Simplified 98.7%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \color{blue}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, -1\right)}\right) \]
12. Add Preprocessing

Alternative 12: 97.6% accurate, 8.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma eps (fma eps (* x 0.16666666666666666) -0.5) (- x))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(eps, (x * 0.16666666666666666), -0.5), -x);
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(eps, Float64(x * 0.16666666666666666), -0.5), Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \cdot \varepsilon} \]

   4. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{6} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{6} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{6} \cdot \varepsilon, -1\right)}\right) \]

   16. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{1}{6}}, -1\right)\right) \]

   17. *-lowering-*.f64 98.2

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.16666666666666666}, -1\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}\right) + -1 \cdot x\right)} \]

    2. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{6} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}, -1 \cdot x\right)} \]

    3. sub-neg N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1 \cdot x\right) \]

    4. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot x\right) \cdot \frac{1}{6}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1 \cdot x\right) \]

    5. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(x \cdot \frac{1}{6}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1 \cdot x\right) \]

    6. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\frac{1}{6} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1 \cdot x\right) \]

    7. metadata-eval N/A

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

    8. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{6} \cdot x, \frac{-1}{2}\right)}, -1 \cdot x\right) \]

    9. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{6}}, \frac{-1}{2}\right), -1 \cdot x\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \frac{1}{6}}, \frac{-1}{2}\right), -1 \cdot x\right) \]

    11. mul-1-neg N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot \frac{1}{6}, \frac{-1}{2}\right), \color{blue}{\mathsf{neg}\left(x\right)}\right) \]

    12. neg-lowering-neg.f64 98.2

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 0.16666666666666666, -0.5\right), \color{blue}{-x}\right) \]

11. Simplified 98.2%

    \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot 0.16666666666666666, -0.5\right), -x\right)} \]
12. Add Preprocessing

Alternative 13: 97.6% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. neg-mul-1 N/A

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

   13. distribute-rgt-out N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sin-lowering-sin.f64 N/A

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

   16. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \cos x, \sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \cdot \varepsilon} \]

   4. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{-1}{2}} + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \]

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{6} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{6} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{6} \cdot \varepsilon, -1\right)}\right) \]

   16. *-commutative N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2}, x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \frac{1}{6}}, -1\right)\right) \]

   17. *-lowering-*.f64 98.2

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.16666666666666666}, -1\right)\right) \]

8. Simplified 98.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5, x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.16666666666666666, -1\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. mul-1-neg N/A

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

    2. neg-lowering-neg.f64 98.2

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

11. Simplified 98.2%

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

Alternative 14: 78.3% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sin-lowering-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 81.0

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

5. Simplified 81.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right) \]
7. Step-by-step derivation

8. Simplified 80.4%

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

Alternative 15: 50.6% 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 52.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

   4. cos-lowering-cos.f64 51.9

      \[\leadsto \color{blue}{\cos \varepsilon} + -1 \]

5. Simplified 51.9%

   \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{1} + -1 \]
7. Step-by-step derivation

8. Simplified 51.9%

   \[\leadsto \color{blue}{1} + -1 \]
9. Step-by-step derivation

   1. metadata-eval 51.9

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

10. Applied egg-rr 51.9%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((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) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :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 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))