2cos (problem 3.3.5)

Percentage Accurate: 52.5% → 99.4%

Time: 16.6s

Alternatives: 14

Speedup: 41.0×

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: 52.5% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. 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 99.8%

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

5. Simplified 99.8%

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \varepsilon\right)\right), \varepsilon\right), \mathsf{*.f64}\left(\left(0 - \sin x\right), \varepsilon\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \varepsilon\right)\right), \varepsilon\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \sin x\right), \varepsilon\right)\right) \]

   13. sin-lowering-sin.f64 99.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \varepsilon\right)\right), \varepsilon\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon\right)\right) \]

7. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. sub0-neg N/A

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

   3. fma-define N/A

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

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

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

   5. sub0-neg N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - \sin x\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \left(\cos x \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin x\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \left(\cos x \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \left(\left(\frac{-1}{2} \cdot \left(\cos x \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \left(\left(\left(\cos x \cdot \varepsilon\right) \cdot \frac{-1}{2}\right) \cdot \varepsilon\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \left(\left(\cos x \cdot \varepsilon\right) \cdot \left(\frac{-1}{2} \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \mathsf{*.f64}\left(\left(\cos x \cdot \varepsilon\right), \left(\frac{-1}{2} \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos x, \varepsilon\right), \left(\frac{-1}{2} \cdot \varepsilon\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \varepsilon\right), \left(\frac{-1}{2} \cdot \varepsilon\right)\right)\right) \]

   13. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(x\right)\right), \varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \varepsilon\right), \mathsf{*.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right) \]

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + x \cdot \left(x \cdot \left(0.25 + x \cdot \left(x \cdot \left(-0.020833333333333332 + \left(x \cdot x\right) \cdot 0.0006944444444444445\right)\right)\right)\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (*
    eps
    (+
     -0.5
     (*
      x
      (*
       x
       (+
        0.25
        (*
         x
         (*
          x
          (+ -0.020833333333333332 (* (* x x) 0.0006944444444444445)))))))))
   (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (x * (x * (0.25 + (x * (x * (-0.020833333333333332 + ((x * x) * 0.0006944444444444445))))))))) - 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 + (x * (x * ((-0.020833333333333332d0) + ((x * x) * 0.0006944444444444445d0))))))))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * (-0.5 + (x * (x * (0.25 + (x * (x * (-0.020833333333333332 + ((x * x) * 0.0006944444444444445))))))))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + (x * (x * (0.25 + (x * (x * (-0.020833333333333332 + ((x * x) * 0.0006944444444444445))))))))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(-0.5 + Float64(x * Float64(x * Float64(0.25 + Float64(x * Float64(x * Float64(-0.020833333333333332 + Float64(Float64(x * x) * 0.0006944444444444445))))))))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + (x * (x * (0.25 + (x * (x * (-0.020833333333333332 + ((x * x) * 0.0006944444444444445))))))))) - sin(x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(-0.5 + N[(x * N[(x * N[(0.25 + N[(x * N[(x * N[(-0.020833333333333332 + N[(N[(x * x), $MachinePrecision] * 0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right) - \frac{1}{2}\right)}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right) + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left({x}^{2} \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \left(x \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \left(\left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right) \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right) \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{1440} \cdot {x}^{2} - \frac{1}{48}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{1440} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{1440} \cdot {x}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{48} + \frac{1}{1440} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{48}, \left(\frac{1}{1440} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

8. Simplified 99.5%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(-0.5 + x \cdot \left(x \cdot \left(0.25 + x \cdot \left(x \cdot \left(-0.020833333333333332 + \left(x \cdot x\right) \cdot 0.0006944444444444445\right)\right)\right)\right)\right)} - \sin x\right) \]
9. Add Preprocessing

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(0.25 + x \cdot \left(x \cdot -0.020833333333333332\right)\right)\right) - \sin x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (* eps (+ -0.5 (* (* x x) (+ 0.25 (* x (* x -0.020833333333333332))))))
   (sin x))))

C

double code(double x, double eps) {
	return eps * ((eps * (-0.5 + ((x * x) * (0.25 + (x * (x * -0.020833333333333332)))))) - 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 + (x * (x * (-0.020833333333333332d0))))))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * (-0.5 + ((x * x) * (0.25 + (x * (x * -0.020833333333333332)))))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * (-0.5 + ((x * x) * (0.25 + (x * (x * -0.020833333333333332)))))) - math.sin(x))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * Float64(-0.5 + Float64(Float64(x * x) * Float64(0.25 + Float64(x * Float64(x * -0.020833333333333332)))))) - sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 + ((x * x) * (0.25 + (x * (x * -0.020833333333333332)))))) - sin(x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(0.25 + N[(x * N[(x * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({x}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right) + \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\frac{-1}{2} + {x}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{4} + \frac{-1}{48} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{-1}{48} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{-1}{48} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(\frac{-1}{48} \cdot x\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(x \cdot \left(\frac{-1}{48} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \left(\frac{-1}{48} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{48}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

   14. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{48}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(x\right)\right)\right) \]

8. Simplified 99.5%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(-0.5 + \left(x \cdot x\right) \cdot \left(0.25 + x \cdot \left(x \cdot -0.020833333333333332\right)\right)\right)} - \sin x\right) \]
9. Add Preprocessing

Alternative 5: 98.8% 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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 99.5%

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

8. Simplified 99.5%

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

Alternative 6: 98.9% 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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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 99.4%

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

8. Simplified 99.4%

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

Alternative 7: 98.4% accurate, 7.6× speedup

Math

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

FPCore

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0

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

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

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

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

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

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

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

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

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

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

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

    12. unpow2 N/A

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

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

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

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

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

    15. *-commutative N/A

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

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

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

    17. unpow2 N/A

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

    18. *-lowering-*.f64 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 8: 98.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0

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

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

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

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

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

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

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

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

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

    11. *-commutative N/A

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

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

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

    13. unpow2 N/A

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

    14. *-lowering-*.f64 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 9: 98.3% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

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) + (eps * 0.25d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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. +-lowering-+.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 98.8%

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

8. Simplified 98.8%

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

Alternative 10: 98.2% 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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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. +-lowering-+.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\varepsilon, \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right)}\right)\right)\right)\right)\right) \]
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 98.8%

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

11. Simplified 98.8%

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

Alternative 11: 97.8% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

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) + (eps * (x * 0.25d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 98.2%

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

8. Simplified 98.2%

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

Alternative 12: 97.8% 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 52.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \cos x\right) - \sin x\right)} \]
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 99.8%

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

5. Simplified 99.8%

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

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 98.2%

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

8. Simplified 98.2%

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

Alternative 13: 79.0% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(0 - x\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (- 0.0 x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (0.0 - x)

Julia

function code(x, eps)
	return Float64(eps * Float64(0.0 - x))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(0.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\mathsf{neg}\left(\varepsilon\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(0 - \color{blue}{\varepsilon}\right)\right) \]

   7. --lowering--.f64 80.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{\_.f64}\left(0, \color{blue}{\varepsilon}\right)\right) \]

5. Simplified 80.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{0}, \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot x\right) \cdot x\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{-1}{6} \cdot x\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot x\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

   8. *-lowering-*.f64 79.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \varepsilon\right)\right) \]

8. Simplified 79.6%

   \[\leadsto \color{blue}{\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\right)} \cdot \left(0 - \varepsilon\right) \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. distribute-rgt-neg-out N/A

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

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

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

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

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

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

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-lowering-*.f64 79.6%

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

10. Applied egg-rr 79.6%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \varepsilon\right)\right) \]
12. Step-by-step derivation

13. Simplified 79.4%

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

14. Final simplification 79.4%

    \[\leadsto \varepsilon \cdot \left(0 - x\right) \]
15. Add Preprocessing

Alternative 14: 51.2% 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 52.9%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   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 52.2%

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

   1. metadata-eval 52.2%

      \[\leadsto 0 \]

10. Applied egg-rr 52.2%

    \[\leadsto \color{blue}{0} \]
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 2024145 
(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)))