2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.9%

Time: 18.3s

Alternatives: 15

Speedup: 205.0×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (/ (sin eps) (cos (+ eps x))) (cos x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/
   (*
    eps
    (+
     1.0
     (*
      eps
      (*
       eps
       (+
        -0.16666666666666666
        (*
         eps
         (*
          eps
          (+
           0.008333333333333333
           (* (* eps eps) -0.0001984126984126984)))))))))
   (cos (+ eps x)))
  (cos x)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return ((eps * (1.0 + (eps * (eps * (-0.16666666666666666 + (eps * (eps * (0.008333333333333333 + ((eps * eps) * -0.0001984126984126984))))))))) / Math.cos((eps + x))) / Math.cos(x);
}

Python

def code(x, eps):
	return ((eps * (1.0 + (eps * (eps * (-0.16666666666666666 + (eps * (eps * (0.008333333333333333 + ((eps * eps) * -0.0001984126984126984))))))))) / math.cos((eps + x))) / math.cos(x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

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

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

   16. *-commutative N/A

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

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

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

   18. unpow2 N/A

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

   19. *-lowering-*.f64 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{\frac{\varepsilon \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-0.16666666666666666 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.008333333333333333 + \left(\varepsilon \cdot \varepsilon\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return ((eps * (1.0 + ((eps * eps) * (-0.16666666666666666 + (0.008333333333333333 * (eps * eps)))))) / Math.cos((eps + x))) / Math.cos(x);
}

Python

def code(x, eps):
	return ((eps * (1.0 + ((eps * eps) * (-0.16666666666666666 + (0.008333333333333333 * (eps * eps)))))) / math.cos((eps + x))) / math.cos(x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \frac{\frac{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \frac{\frac{\varepsilon \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666\right)\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\varepsilon}, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]
6. Step-by-step derivation

7. Simplified 98.7%

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

8. Final simplification 98.7%

   \[\leadsto \frac{\frac{\varepsilon}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow (tan x) 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(tan(x), 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (tan(x) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(Math.tan(x), 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(math.tan(x), 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (tan(x) ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

      \[\leadsto \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + \color{blue}{1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

      \[\leadsto \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + \varepsilon \]

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

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

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

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. times-frac N/A

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

   9. tan-quot N/A

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

   10. tan-quot N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\tan x \cdot \tan x\right), \varepsilon\right), \varepsilon\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({\tan x}^{2}\right), \varepsilon\right), \varepsilon\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\tan x, 2\right), \varepsilon\right), \varepsilon\right) \]

   13. tan-lowering-tan.f64 98.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), \varepsilon\right), \varepsilon\right) \]

7. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

8. Final simplification 98.4%

   \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
9. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (cos(x) ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps / Math.pow(Math.cos(x), 2.0);
}

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. sin-diff N/A

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

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

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

   9. associate--l+ N/A

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \cos \left(x + \varepsilon\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\left(x + \varepsilon\right)\right)\right), \cos x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \cos x\right) \]

   14. cos-lowering-cos.f64 63.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\varepsilon, x\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(x, \varepsilon\right)\right)\right), \mathsf{cos.f64}\left(x\right)\right) \]

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \color{blue}{\left({\cos x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{pow.f64}\left(\cos x, \color{blue}{2}\right)\right) \]

   3. cos-lowering-cos.f64 98.4%

      \[\leadsto \mathsf{/.f64}\left(\varepsilon, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(x\right), 2\right)\right) \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 8: 98.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.37777777777777777 + x \cdot \left(x \cdot 0.19682539682539682\right)\right)\right)\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   (+
    1.0
    (*
     x
     (*
      x
      (+
       0.6666666666666666
       (* (* x x) (+ 0.37777777777777777 (* x (* x 0.19682539682539682))))))))
   (* eps (* x x)))))

C

double code(double x, double eps) {
	return eps + ((1.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.37777777777777777 + (x * (x * 0.19682539682539682)))))))) * (eps * (x * x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((1.0d0 + (x * (x * (0.6666666666666666d0 + ((x * x) * (0.37777777777777777d0 + (x * (x * 0.19682539682539682d0)))))))) * (eps * (x * x)))
end function

Java

public static double code(double x, double eps) {
	return eps + ((1.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.37777777777777777 + (x * (x * 0.19682539682539682)))))))) * (eps * (x * x)));
}

Python

def code(x, eps):
	return eps + ((1.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.37777777777777777 + (x * (x * 0.19682539682539682)))))))) * (eps * (x * x)))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(1.0 + Float64(x * Float64(x * Float64(0.6666666666666666 + Float64(Float64(x * x) * Float64(0.37777777777777777 + Float64(x * Float64(x * 0.19682539682539682)))))))) * Float64(eps * Float64(x * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((1.0 + (x * (x * (0.6666666666666666 + ((x * x) * (0.37777777777777777 + (x * (x * 0.19682539682539682)))))))) * (eps * (x * x)));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(1.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.37777777777777777 + N[(x * N[(x * 0.19682539682539682), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)}\right)\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2}{3}} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2}{3}} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left({x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{17}{45}} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{17}{45}, \color{blue}{\left(\frac{62}{315} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{17}{45}, \left({x}^{2} \cdot \color{blue}{\frac{62}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{17}{45}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{62}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{17}{45}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{62}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 97.7%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{17}{45}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{62}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.7%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.37777777777777777 + \left(x \cdot x\right) \cdot 0.19682539682539682\right)\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{17}{45} + \left(x \cdot x\right) \cdot \frac{62}{315}\right)\right)\right)\right) + \color{blue}{1}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{17}{45} + \left(x \cdot x\right) \cdot \frac{62}{315}\right)\right)\right)\right)\right) \cdot \varepsilon + \color{blue}{1 \cdot \varepsilon} \]

   3. *-lft-identity N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{17}{45} + \left(x \cdot x\right) \cdot \frac{62}{315}\right)\right)\right)\right)\right) \cdot \varepsilon + \varepsilon \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{17}{45} + \left(x \cdot x\right) \cdot \frac{62}{315}\right)\right)\right)\right)\right) \cdot \varepsilon\right), \color{blue}{\varepsilon}\right) \]

10. Applied egg-rr 97.7%

    \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.37777777777777777 + x \cdot \left(x \cdot 0.19682539682539682\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \varepsilon\right) + \varepsilon} \]

11. Final simplification 97.7%

    \[\leadsto \varepsilon + \left(1 + x \cdot \left(x \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.37777777777777777 + x \cdot \left(x \cdot 0.19682539682539682\right)\right)\right)\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right) \]
12. Add Preprocessing

Alternative 9: 98.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \left(x \cdot x\right) \cdot \left(\varepsilon + \left(x \cdot x\right) \cdot \left(0.37777777777777777 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   (* x x)
   (+
    eps
    (*
     (* x x)
     (+
      (* 0.37777777777777777 (* eps (* x x)))
      (* eps 0.6666666666666666)))))))

C

double code(double x, double eps) {
	return eps + ((x * x) * (eps + ((x * x) * ((0.37777777777777777 * (eps * (x * x))) + (eps * 0.6666666666666666)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((x * x) * (eps + ((x * x) * ((0.37777777777777777d0 * (eps * (x * x))) + (eps * 0.6666666666666666d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps + ((x * x) * (eps + ((x * x) * ((0.37777777777777777 * (eps * (x * x))) + (eps * 0.6666666666666666)))));
}

Python

def code(x, eps):
	return eps + ((x * x) * (eps + ((x * x) * ((0.37777777777777777 * (eps * (x * x))) + (eps * 0.6666666666666666)))))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(x * x) * Float64(eps + Float64(Float64(x * x) * Float64(Float64(0.37777777777777777 * Float64(eps * Float64(x * x))) + Float64(eps * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((x * x) * (eps + ((x * x) * ((0.37777777777777777 * (eps * (x * x))) + (eps * 0.6666666666666666)))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(x * x), $MachinePrecision] * N[(eps + N[(N[(x * x), $MachinePrecision] * N[(N[(0.37777777777777777 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left({x}^{2} \cdot \left(\varepsilon + {x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\varepsilon + {x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\varepsilon} + {x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\varepsilon} + {x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left({x}^{2} \cdot \left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{2}{3} \cdot \varepsilon\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right)} + \frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right)} + \frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(\frac{17}{45} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right), \color{blue}{\left(\frac{2}{3} \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(\left(\varepsilon \cdot {x}^{2}\right) \cdot \frac{17}{45}\right), \left(\color{blue}{\frac{2}{3}} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot {x}^{2}\right), \frac{17}{45}\right), \left(\color{blue}{\frac{2}{3}} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2} \cdot \varepsilon\right), \frac{17}{45}\right), \left(\frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \varepsilon\right), \frac{17}{45}\right), \left(\frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \varepsilon\right), \frac{17}{45}\right), \left(\frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{17}{45}\right), \left(\frac{2}{3} \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{17}{45}\right), \left(\varepsilon \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 97.7%

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \varepsilon\right), \frac{17}{45}\right), \mathsf{*.f64}\left(\varepsilon, \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.7%

   \[\leadsto \color{blue}{\varepsilon + \left(x \cdot x\right) \cdot \left(\varepsilon + \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 0.37777777777777777 + \varepsilon \cdot 0.6666666666666666\right)\right)} \]

9. Final simplification 97.7%

   \[\leadsto \varepsilon + \left(x \cdot x\right) \cdot \left(\varepsilon + \left(x \cdot x\right) \cdot \left(0.37777777777777777 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right) + \varepsilon \cdot 0.6666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 10: 98.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   eps
   (*
    x
    (*
     x
     (+
      1.0
      (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.37777777777777777)))))))))

C

double code(double x, double eps) {
	return eps + (eps * (x * (x * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (x * (x * (1.0d0 + ((x * x) * (0.6666666666666666d0 + ((x * x) * 0.37777777777777777d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (x * (x * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777)))))));
}

Python

def code(x, eps):
	return eps + (eps * (x * (x * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777)))))))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64(x * Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.37777777777777777))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x * (x * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777)))))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(x * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.37777777777777777), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2}{3}} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2}{3}} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{17}{45} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \color{blue}{\frac{17}{45}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{17}{45}}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{17}{45}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 97.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{45}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.6%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \frac{17}{45}\right)\right) + \color{blue}{1}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \frac{17}{45}\right)\right)\right) + \color{blue}{\varepsilon \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \frac{17}{45}\right)\right)\right) + \varepsilon \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \frac{17}{45}\right)\right)\right)\right), \color{blue}{\varepsilon}\right) \]

10. Applied egg-rr 97.7%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right)\right) + \varepsilon} \]

11. Final simplification 97.7%

    \[\leadsto \varepsilon + \varepsilon \cdot \left(x \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 98.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    (* x x)
    (+
     1.0
     (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.37777777777777777))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((x * x) * (1.0d0 + ((x * x) * (0.6666666666666666d0 + ((x * x) * 0.37777777777777777d0))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777))))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((x * x) * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.37777777777777777)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((x * x) * (1.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.37777777777777777))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.37777777777777777), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{2}{3}} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{2}{3}} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{17}{45} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \color{blue}{\frac{17}{45}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{17}{45}}\right)\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{17}{45}\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 97.6%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{17}{45}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.6%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.37777777777777777\right)\right)\right)} \]
9. Add Preprocessing

Alternative 12: 98.3% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon + x \cdot \left(x \cdot \left(\varepsilon + \varepsilon \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* x (* x (+ eps (* eps (* 0.6666666666666666 (* x x))))))))

C

double code(double x, double eps) {
	return eps + (x * (x * (eps + (eps * (0.6666666666666666 * (x * x))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (x * (x * (eps + (eps * (0.6666666666666666d0 * (x * x))))))
end function

Java

public static double code(double x, double eps) {
	return eps + (x * (x * (eps + (eps * (0.6666666666666666 * (x * x))))));
}

Python

def code(x, eps):
	return eps + (x * (x * (eps + (eps * (0.6666666666666666 * (x * x))))))

Julia

function code(x, eps)
	return Float64(eps + Float64(x * Float64(x * Float64(eps + Float64(eps * Float64(0.6666666666666666 * Float64(x * x)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (x * (x * (eps + (eps * (0.6666666666666666 * (x * x))))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(x * N[(x * N[(eps + N[(eps * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left({x}^{2} \cdot \left(\varepsilon + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\varepsilon} + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\left(x \cdot \left(\varepsilon + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\varepsilon + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon + \frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(\frac{2}{3} \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \left(\left(\varepsilon \cdot {x}^{2}\right) \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \left(\varepsilon \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{3}\right)}\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \left(\varepsilon \cdot \left(\frac{2}{3} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{2}{3} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \left({x}^{2} \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 97.5%

       \[\leadsto \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.5%

   \[\leadsto \color{blue}{\varepsilon + x \cdot \left(x \cdot \left(\varepsilon + \varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)\right)} \]

9. Final simplification 97.5%

   \[\leadsto \varepsilon + x \cdot \left(x \cdot \left(\varepsilon + \varepsilon \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 13: 98.3% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* (* x x) (+ 1.0 (* 0.6666666666666666 (* x x)))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (1.0 + (0.6666666666666666 * (x * x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((x * x) * (1.0d0 + (0.6666666666666666d0 * (x * x)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((x * x) * (1.0 + (0.6666666666666666 * (x * x)))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((x * x) * (1.0 + (0.6666666666666666 * (x * x)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(0.6666666666666666 * Float64(x * x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((x * x) * (1.0 + (0.6666666666666666 * (x * x)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + {x}^{2} \cdot \left(1 + \frac{2}{3} \cdot {x}^{2}\right)\right)}\right) \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{2}{3} \cdot {x}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(1 + \frac{2}{3} \cdot {x}^{2}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{1} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{1} + \frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{2}{3} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 97.5%

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right)\right)\right)\right)\right) \]

8. Simplified 97.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right)} \]

9. Final simplification 97.5%

   \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \varepsilon + {x}^{2} \cdot \color{blue}{\varepsilon} \]

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot x\right) \]
10. Add Preprocessing

Alternative 15: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

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

8. Simplified 96.9%

   \[\leadsto \color{blue}{\varepsilon} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024164 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  (- (tan (+ x eps)) (tan x)))