2cos (problem 3.3.5)

Percentage Accurate: 52.2% → 99.5%

Time: 36.2s

Alternatives: 14

Speedup: 29.3×

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: 52.2% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return fma(Float64(sin(x) * Float64(Float64(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[(N[(eps * N[(eps * 0.16666666666666666), $MachinePrecision]), $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 53.3%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \left(\color{blue}{\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \color{blue}{\varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + -1 \cdot \color{blue}{\sin x}\right)\right)\right) \]

5. Simplified 99.7%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \color{blue}{\varepsilon}, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

   15. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right) \]

7. Applied egg-rr 99.9%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((sin(x) * ((eps * (eps * 0.16666666666666666d0)) + (-1.0d0))) + (eps * ((-0.5d0) * cos(x))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.sin(x) * ((eps * (eps * 0.16666666666666666)) + -1.0)) + (eps * (-0.5 * Math.cos(x))));
}

Python

def code(x, eps):
	return eps * ((math.sin(x) * ((eps * (eps * 0.16666666666666666)) + -1.0)) + (eps * (-0.5 * math.cos(x))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((sin(x) * ((eps * (eps * 0.16666666666666666)) + -1.0)) + (eps * (-0.5 * cos(x))));
end

Wolfram

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

Derivation

1. Initial program 53.3%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \left(\color{blue}{\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \color{blue}{\varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + -1 \cdot \color{blue}{\sin x}\right)\right)\right) \]

5. Simplified 99.7%

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

6. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.16666666666666666\right) + -1\right) + \varepsilon \cdot \left(-0.5 \cdot \cos x\right)\right) \]
7. Add Preprocessing

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * (0.5d0 + ((eps * eps) * (-0.020833333333333332d0)))) * sin(((eps + (x * 2.0d0)) / 2.0d0))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return ((eps * (0.5 + ((eps * eps) * -0.020833333333333332))) * Math.sin(((eps + (x * 2.0)) / 2.0))) * -2.0;
}

Python

def code(x, eps):
	return ((eps * (0.5 + ((eps * eps) * -0.020833333333333332))) * math.sin(((eps + (x * 2.0)) / 2.0))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * (0.5 + ((eps * eps) * -0.020833333333333332))) * sin(((eps + (x * 2.0)) / 2.0))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \left({\varepsilon}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

   5. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \left(\left(\varepsilon \cdot \left(0.5 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.020833333333333332\right)\right) \cdot \sin \left(\frac{\varepsilon + x \cdot 2}{2}\right)\right) \cdot -2 \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-2.0d0) * (sin(((eps + (x * 2.0d0)) / 2.0d0)) * (eps * 0.5d0))
end function

Java

public static double code(double x, double eps) {
	return -2.0 * (Math.sin(((eps + (x * 2.0)) / 2.0)) * (eps * 0.5));
}

Python

def code(x, eps):
	return -2.0 * (math.sin(((eps + (x * 2.0)) / 2.0)) * (eps * 0.5))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin(((eps + (x * 2.0)) / 2.0)) * (eps * 0.5));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right), \color{blue}{-2}\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \frac{1}{2}\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

   2. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{2}\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, 2\right)\right), 2\right)\right)\right), -2\right) \]

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon + x \cdot 2}{2}\right) \cdot \left(\varepsilon \cdot 0.5\right)\right) \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (- (* eps (+ -0.5 (* x (* x 0.25)))) (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (x * (x * 0.25)))) - sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((-0.5d0) + (x * (x * 0.25d0)))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (x * (x * 0.25)))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + (x * (x * 0.25)))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(-0.5 + Float64(x * Float64(x * 0.25)))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + (x * (x * 0.25)))) - sin(x));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)}, \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot \varepsilon\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\varepsilon \cdot \frac{1}{4}\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{4} \cdot {x}^{2}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   18. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

8. Simplified 99.3%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right)} - \sin x\right) \]
9. Add Preprocessing

Alternative 6: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) - math.sin(x))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) - sin(x));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)}, \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   2. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

8. Simplified 99.2%

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

Alternative 7: 98.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((-0.5d0) + (x * (x * 0.25d0)))) + (x * ((-1.0d0) - (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.008333333333333333d0 + ((x * x) * (-0.0001984126984126984d0))))))))))
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + (x * (x * 0.25)))) + (x * (-1.0 - (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + (x * (x * 0.25)))) + (x * (-1.0 - (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((x * x) * -0.0001984126984126984)))))))));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)}, \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot \varepsilon\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\varepsilon \cdot \frac{1}{4}\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{4} \cdot {x}^{2}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   18. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

8. Simplified 99.3%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right)} - \sin x\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right)\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{120}} + \frac{-1}{5040} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    18. *-lowering-*.f64 98.9%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.9%

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

12. Final simplification 98.9%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right) + x \cdot \left(-1 - x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 98.2% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((-0.5d0) + (x * (x * 0.25d0)))) + (x * ((-1.0d0) - (x * (x * ((-0.16666666666666666d0) + ((x * x) * 0.008333333333333333d0)))))))
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + (x * (x * 0.25)))) + (x * (-1.0 - (x * (x * (-0.16666666666666666 + ((x * x) * 0.008333333333333333)))))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + (x * (x * 0.25)))) + (x * (-1.0 - (x * (x * (-0.16666666666666666 + ((x * x) * 0.008333333333333333)))))));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)}, \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot \varepsilon\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \left(\varepsilon \cdot \frac{1}{4}\right) \cdot {x}^{2}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2} + \varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{x}\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{1}{4} \cdot {x}^{2} + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + \frac{1}{4} \cdot {x}^{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{4} \cdot {x}^{2}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   18. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

8. Simplified 99.3%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right)} - \sin x\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right)}\right)\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {x}^{2}} - \frac{1}{6}\right)\right)\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

    14. *-lowering-*.f64 98.8%

        \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot 0.25\right)\right) + x \cdot \left(-1 - x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 9: 98.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((-1.0d0) + (x * ((eps * 0.25d0) + (x * 0.16666666666666666d0))))))
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * (-1.0 + (x * ((eps * 0.25) + (x * 0.16666666666666666))))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * (-1.0 + (x * ((eps * 0.25) + (x * 0.16666666666666666))))));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + -1\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \varepsilon + \color{blue}{\frac{1}{6} \cdot x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{4} \cdot \varepsilon\right), \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{1}{4}\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 98.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 98.6%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5 + x \cdot \left(-1 + x \cdot \left(\varepsilon \cdot 0.25 + x \cdot 0.16666666666666666\right)\right)\right)} \]
9. Add Preprocessing

Alternative 10: 97.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((-1.0d0) + (x * (x * 0.16666666666666666d0)))))
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * (-1.0 + (x * (x * 0.16666666666666666)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * (-1.0 + (x * (x * 0.16666666666666666)))));
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + -1\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \left(-1 + \color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \varepsilon + \color{blue}{\frac{1}{6} \cdot x}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{4} \cdot \varepsilon\right), \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon \cdot \frac{1}{4}\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \left(\color{blue}{\frac{1}{6}} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 98.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{1}{4}\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 98.6%

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

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

    2. *-lowering-*.f64 98.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

11. Simplified 98.6%

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

Alternative 11: 54.1% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-147}:\\ \;\;\;\;0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.8e-147) (* 0.5 (* x x)) (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e-147) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.8d-147)) then
        tmp = 0.5d0 * (x * x)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e-147) {
		tmp = 0.5 * (x * x);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.8e-147:
		tmp = 0.5 * (x * x)
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.8e-147)
		tmp = Float64(0.5 * Float64(x * x));
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.8e-147)
		tmp = 0.5 * (x * x);
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.8e-147], N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000006e-147

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 7.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 12.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   8. Simplified 12.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]

   if -1.80000000000000006e-147 < x

   1. Initial program 66.9%

      \[\cos \left(x + \varepsilon\right) - \cos x \]
   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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \left(\color{blue}{\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \color{blue}{\varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + -1 \cdot \color{blue}{\sin x}\right)\right)\right) \]

   5. Simplified 99.8%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. fma-define N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(\left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \color{blue}{\varepsilon}, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

      15. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right) \]

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({\varepsilon}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right) \]

      3. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 97.5% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) - x)
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * -0.5) - x)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) - x);
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in eps around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \cos x\right) \cdot \frac{-1}{2} - \sin \color{blue}{x}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\cos x \cdot \frac{-1}{2}\right) - \sin \color{blue}{x}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) - \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\sin x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \sin \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \sin x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \sin x\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(-1 \cdot x + \frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{-1 \cdot x}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \varepsilon + \left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \varepsilon - \color{blue}{x}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \varepsilon\right), \color{blue}{x}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\left(\varepsilon \cdot \frac{-1}{2}\right), x\right)\right) \]

   14. *-lowering-*.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), x\right)\right) \]

8. Simplified 98.1%

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

Alternative 13: 52.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (-0.5d0) * (eps * eps)
end function

Java

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

Python

def code(x, eps):
	return -0.5 * (eps * eps)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -0.5 * (eps * eps);
end

Wolfram

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

Derivation

1. Initial program 53.3%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)}\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right) + \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right), \color{blue}{\left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} \cdot \cos x\right)\right), \left(\color{blue}{\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \color{blue}{\varepsilon} + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\varepsilon \cdot \left(\left(\frac{1}{6} \cdot \varepsilon\right) \cdot \sin x\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + \left(\mathsf{neg}\left(\color{blue}{\sin x}\right)\right)\right)\right)\right) \]

   13. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right)\right), \left(\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \varepsilon\right)\right) \cdot \sin x + -1 \cdot \color{blue}{\sin x}\right)\right)\right) \]

5. Simplified 99.7%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. fma-define N/A

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

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

      \[\leadsto \mathsf{fma.f64}\left(\left(\sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \color{blue}{\varepsilon}, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right) + -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x\right)\right) \cdot \varepsilon\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\left(\frac{-1}{2} \cdot \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \cos x\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \left(\varepsilon \cdot \varepsilon\right)\right)\right) \]

   15. *-lowering-*.f64 99.9%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \frac{1}{6}\right)\right), -1\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(x\right)\right), \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right) \]

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({\varepsilon}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right) \]

   3. *-lowering-*.f64 53.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\varepsilon}\right)\right) \]

10. Simplified 53.4%

    \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
11. Add Preprocessing

Alternative 14: 50.7% accurate, 205.0× 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 53.3%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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

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

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + -1 \]

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

      \[\leadsto \mathsf{+.f64}\left(\cos \varepsilon, \color{blue}{-1}\right) \]

   4. cos-lowering-cos.f64 52.3%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\varepsilon\right), -1\right) \]

5. Simplified 52.3%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{1}, -1\right) \]
7. Step-by-step derivation

8. Simplified 52.3%

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

   1. metadata-eval 52.3%

      \[\leadsto 0 \]

10. Applied egg-rr 52.3%

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

Developer Target 1: 99.7% accurate, 1.0× 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]

Reproduce

herbie shell --seed 2024141 
(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))))

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