2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 17.3s

Alternatives: 11

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: 62.5% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \varepsilon + \varepsilon \cdot \left(t\_0 + \varepsilon \cdot \left(\tan x + \left({\tan x}^{3} + \varepsilon \cdot \left(\left(\left(\left(t\_0 + {\tan x}^{4}\right) - 0.16666666666666666\right) - t\_0 \cdot 0.16666666666666666\right) - \left(-0.5 + t\_0 \cdot -0.5\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (+
    eps
    (*
     eps
     (+
      t_0
      (*
       eps
       (+
        (tan x)
        (+
         (pow (tan x) 3.0)
         (*
          eps
          (-
           (-
            (- (+ t_0 (pow (tan x) 4.0)) 0.16666666666666666)
            (* t_0 0.16666666666666666))
           (+ -0.5 (* t_0 -0.5))))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return eps + (eps * (t_0 + (eps * (tan(x) + (pow(tan(x), 3.0) + (eps * ((((t_0 + pow(tan(x), 4.0)) - 0.16666666666666666) - (t_0 * 0.16666666666666666)) - (-0.5 + (t_0 * -0.5)))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = eps + (eps * (t_0 + (eps * (tan(x) + ((tan(x) ** 3.0d0) + (eps * ((((t_0 + (tan(x) ** 4.0d0)) - 0.16666666666666666d0) - (t_0 * 0.16666666666666666d0)) - ((-0.5d0) + (t_0 * (-0.5d0))))))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return eps + (eps * (t_0 + (eps * (Math.tan(x) + (Math.pow(Math.tan(x), 3.0) + (eps * ((((t_0 + Math.pow(Math.tan(x), 4.0)) - 0.16666666666666666) - (t_0 * 0.16666666666666666)) - (-0.5 + (t_0 * -0.5)))))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.tan(x), 2.0)
	return eps + (eps * (t_0 + (eps * (math.tan(x) + (math.pow(math.tan(x), 3.0) + (eps * ((((t_0 + math.pow(math.tan(x), 4.0)) - 0.16666666666666666) - (t_0 * 0.16666666666666666)) - (-0.5 + (t_0 * -0.5)))))))))

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(eps + Float64(eps * Float64(t_0 + Float64(eps * Float64(tan(x) + Float64((tan(x) ^ 3.0) + Float64(eps * Float64(Float64(Float64(Float64(t_0 + (tan(x) ^ 4.0)) - 0.16666666666666666) - Float64(t_0 * 0.16666666666666666)) - Float64(-0.5 + Float64(t_0 * -0.5))))))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) ^ 2.0;
	tmp = eps + (eps * (t_0 + (eps * (tan(x) + ((tan(x) ^ 3.0) + (eps * ((((t_0 + (tan(x) ^ 4.0)) - 0.16666666666666666) - (t_0 * 0.16666666666666666)) - (-0.5 + (t_0 * -0.5)))))))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps + N[(eps * N[(t$95$0 + N[(eps * N[(N[Tan[x], $MachinePrecision] + N[(N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision] + N[(eps * N[(N[(N[(N[(t$95$0 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(-0.5 + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \varepsilon \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{{\tan x}^{-2}} + \varepsilon \cdot \left(\left(\tan x + \frac{1}{{\tan x}^{-3}}\right) - \varepsilon \cdot \left(\left(0.16666666666666666 + \frac{0.16666666666666666}{{\tan x}^{-2}}\right) - \left(\frac{1 + \frac{1}{{\tan x}^{-2}}}{{\tan x}^{-2}} - \left(-0.5 + \frac{-0.5}{{\tan x}^{-2}}\right)\right)\right)\right)\right) + \varepsilon} \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left({\tan x}^{2} + \varepsilon \cdot \left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(-0.5 + {\tan x}^{2} \cdot -0.5\right) + \left({\tan x}^{2} \cdot 0.16666666666666666 + \left(0.16666666666666666 - {\tan x}^{2} \cdot \left(1 + {\tan x}^{2}\right)\right)\right)\right)\right)\right) \cdot \varepsilon} + \varepsilon \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), \left(\left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(\frac{-1}{2} + {\tan x}^{2} \cdot \frac{-1}{2}\right) + \left({\tan x}^{2} \cdot \frac{1}{6} + \left(\frac{1}{6} - {\tan x}^{2} \cdot \left(1 + {\tan x}^{2}\right)\right)\right)\right)\right) \cdot \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), \mathsf{*.f64}\left(\left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(\frac{-1}{2} + {\tan x}^{2} \cdot \frac{-1}{2}\right) + \left({\tan x}^{2} \cdot \frac{1}{6} + \left(\frac{1}{6} - {\tan x}^{2} \cdot \left(1 + {\tan x}^{2}\right)\right)\right)\right)\right), \varepsilon\right)\right), \varepsilon\right), \varepsilon\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \left({\tan x}^{2} + \color{blue}{\left(\tan x + \left({\tan x}^{3} - \varepsilon \cdot \left(\left(-0.5 + {\tan x}^{2} \cdot -0.5\right) + \left({\tan x}^{2} \cdot 0.16666666666666666 + \left(0.16666666666666666 - \left({\tan x}^{2} + {\tan x}^{4}\right)\right)\right)\right)\right)\right) \cdot \varepsilon}\right) \cdot \varepsilon + \varepsilon \]

12. Final simplification 100.0%

    \[\leadsto \varepsilon + \varepsilon \cdot \left({\tan x}^{2} + \varepsilon \cdot \left(\tan x + \left({\tan x}^{3} + \varepsilon \cdot \left(\left(\left(\left({\tan x}^{2} + {\tan x}^{4}\right) - 0.16666666666666666\right) - {\tan x}^{2} \cdot 0.16666666666666666\right) - \left(-0.5 + {\tan x}^{2} \cdot -0.5\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \left({\tan x}^{2} - \varepsilon \cdot \left(\varepsilon \cdot -0.3333333333333333 - \left(\tan x + {\tan x}^{3}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  eps
  (*
   eps
   (-
    (pow (tan x) 2.0)
    (* eps (- (* eps -0.3333333333333333) (+ (tan x) (pow (tan x) 3.0))))))))

C

double code(double x, double eps) {
	return eps + (eps * (pow(tan(x), 2.0) - (eps * ((eps * -0.3333333333333333) - (tan(x) + pow(tan(x), 3.0))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * ((tan(x) ** 2.0d0) - (eps * ((eps * (-0.3333333333333333d0)) - (tan(x) + (tan(x) ** 3.0d0))))))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (Math.pow(Math.tan(x), 2.0) - (eps * ((eps * -0.3333333333333333) - (Math.tan(x) + Math.pow(Math.tan(x), 3.0))))));
}

Python

def code(x, eps):
	return eps + (eps * (math.pow(math.tan(x), 2.0) - (eps * ((eps * -0.3333333333333333) - (math.tan(x) + math.pow(math.tan(x), 3.0))))))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64((tan(x) ^ 2.0) - Float64(eps * Float64(Float64(eps * -0.3333333333333333) - Float64(tan(x) + (tan(x) ^ 3.0)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * ((tan(x) ^ 2.0) - (eps * ((eps * -0.3333333333333333) - (tan(x) + (tan(x) ^ 3.0))))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - N[(eps * N[(N[(eps * -0.3333333333333333), $MachinePrecision] - N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \varepsilon \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{1}{{\tan x}^{-2}} + \varepsilon \cdot \left(\left(\tan x + \frac{1}{{\tan x}^{-3}}\right) - \varepsilon \cdot \left(\left(0.16666666666666666 + \frac{0.16666666666666666}{{\tan x}^{-2}}\right) - \left(\frac{1 + \frac{1}{{\tan x}^{-2}}}{{\tan x}^{-2}} - \left(-0.5 + \frac{-0.5}{{\tan x}^{-2}}\right)\right)\right)\right)\right) + \varepsilon} \]

9. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left({\tan x}^{2} + \varepsilon \cdot \left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \left(\left(-0.5 + {\tan x}^{2} \cdot -0.5\right) + \left({\tan x}^{2} \cdot 0.16666666666666666 + \left(0.16666666666666666 - {\tan x}^{2} \cdot \left(1 + {\tan x}^{2}\right)\right)\right)\right)\right)\right) \cdot \varepsilon} + \varepsilon \]

10. Taylor expanded in x around 0

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

12. Simplified 99.9%

    \[\leadsto \left({\tan x}^{2} + \varepsilon \cdot \left(\left(\tan x + {\tan x}^{3}\right) - \varepsilon \cdot \color{blue}{-0.3333333333333333}\right)\right) \cdot \varepsilon + \varepsilon \]

13. Final simplification 99.9%

    \[\leadsto \varepsilon + \varepsilon \cdot \left({\tan x}^{2} - \varepsilon \cdot \left(\varepsilon \cdot -0.3333333333333333 - \left(\tan x + {\tan x}^{3}\right)\right)\right) \]
14. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. frac-times N/A

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

   7. tan-quot N/A

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

   8. tan-quot N/A

      \[\leadsto \left(\tan x \cdot \tan x\right) \cdot \varepsilon + \varepsilon \]

   9. pow2 N/A

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

   10. metadata-eval N/A

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

   11. pow-flip N/A

       \[\leadsto \frac{1}{{\tan x}^{-2}} \cdot \varepsilon + \varepsilon \]

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

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

   13. *-commutative N/A

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

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

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

   15. pow-flip N/A

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

   16. metadata-eval N/A

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

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

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

   18. tan-lowering-tan.f64 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 98.4% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

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. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.5%

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

8. Simplified 99.5%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 98.4% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

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. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 6: 98.3% 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 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

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. *-lowering-*.f64 N/A

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

   13. unpow2 N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 7: 98.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

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 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 8: 98.3% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

5. Simplified 99.8%

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

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. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 9: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \varepsilon \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{4}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{4}{3}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{4}{3}}\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{4}{3}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{4}{3}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right)\right)} \]

9. Taylor expanded in eps around 0

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

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

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 99.4%

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

11. Simplified 99.4%

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. *-lft-identity N/A

       \[\leadsto \left(x \cdot x\right) \cdot \varepsilon + \varepsilon \]

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

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

    5. *-commutative N/A

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

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

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

    7. *-lowering-*.f64 99.4%

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

13. Applied egg-rr 99.4%

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

14. Final simplification 99.4%

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

Alternative 10: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \varepsilon \cdot \left(\left(-0.5 + \frac{-0.5 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 - \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {\varepsilon}^{2}\right)\right), \left(\color{blue}{x} \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left({\varepsilon}^{2}\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \color{blue}{\left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{4}{3} \cdot {\varepsilon}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{4}{3}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{4}{3}}\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{4}{3}\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{4}{3}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right)\right)\right)} \]

9. Taylor expanded in eps around 0

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

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

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 11: 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 61.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. cos-lowering-cos.f64 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in eps around 0

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

8. Simplified 98.9%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024152 
(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)))