2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.6%

Time: 34.2s

Alternatives: 12

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: 62.7% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{-2}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, 1\right)\\ t_3 := {\cos x}^{2}\\ \varepsilon \cdot \left(\left(1 + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(-\varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, t\_0 \cdot \frac{t\_2}{t\_3}, \mathsf{fma}\left(0.16666666666666666, t\_0 \cdot t\_1, t\_2 \cdot -0.5\right)\right), \sin x \cdot \frac{t\_2}{\cos x}\right)\right)\right)\right) + \frac{t\_0}{t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) -2.0))
        (t_2 (fma t_0 t_1 1.0))
        (t_3 (pow (cos x) 2.0)))
   (*
    eps
    (+
     (+
      1.0
      (log
       (+
        1.0
        (expm1
         (*
          eps
          (fma
           (- eps)
           (+
            0.16666666666666666
            (fma
             -1.0
             (* t_0 (/ t_2 t_3))
             (fma 0.16666666666666666 (* t_0 t_1) (* t_2 -0.5))))
           (* (sin x) (/ t_2 (cos x)))))))))
     (/ t_0 t_3)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), -2.0);
	double t_2 = fma(t_0, t_1, 1.0);
	double t_3 = pow(cos(x), 2.0);
	return eps * ((1.0 + log((1.0 + expm1((eps * fma(-eps, (0.16666666666666666 + fma(-1.0, (t_0 * (t_2 / t_3)), fma(0.16666666666666666, (t_0 * t_1), (t_2 * -0.5)))), (sin(x) * (t_2 / cos(x))))))))) + (t_0 / t_3));
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ -2.0
	t_2 = fma(t_0, t_1, 1.0)
	t_3 = cos(x) ^ 2.0
	return Float64(eps * Float64(Float64(1.0 + log(Float64(1.0 + expm1(Float64(eps * fma(Float64(-eps), Float64(0.16666666666666666 + fma(-1.0, Float64(t_0 * Float64(t_2 / t_3)), fma(0.16666666666666666, Float64(t_0 * t_1), Float64(t_2 * -0.5)))), Float64(sin(x) * Float64(t_2 / cos(x))))))))) + Float64(t_0 / t_3)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[Log[N[(1.0 + N[(Exp[N[(eps * N[((-eps) * N[(0.16666666666666666 + N[(-1.0 * N[(t$95$0 * N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$2 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(-1 \cdot \varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, {\sin x}^{2} \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right)}{{\cos x}^{2}}, \mathsf{fma}\left(0.16666666666666666, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot -0.5\right)\right), \sin x \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right)}{\cos x}\right)\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

6. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \log \left(1 + \mathsf{expm1}\left(\varepsilon \cdot \mathsf{fma}\left(-\varepsilon, 0.16666666666666666 + \mathsf{fma}\left(-1, {\sin x}^{2} \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right)}{{\cos x}^{2}}, \mathsf{fma}\left(0.16666666666666666, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right) \cdot -0.5\right)\right), \sin x \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, {\cos x}^{-2}, 1\right)}{\cos x}\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := {\cos x}^{2}\\ t_2 := {\sin x}^{2}\\ t_3 := \frac{t\_2}{t\_1}\\ t_4 := 1 + t\_3\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{t\_2 \cdot t\_4}{t\_1} - \left(-0.5 \cdot t\_4 + 0.16666666666666666 \cdot t\_3\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot t\_4}{\cos x}\right)\right) + \frac{0.5 - \frac{t\_0}{2}}{\frac{1 + t\_0}{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0)))
        (t_1 (pow (cos x) 2.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (/ t_2 t_1))
        (t_4 (+ 1.0 t_3)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (*
         eps
         (-
          (- (/ (* t_2 t_4) t_1) (+ (* -0.5 t_4) (* 0.16666666666666666 t_3)))
          0.16666666666666666))
        (/ (* (sin x) t_4) (cos x)))))
     (/ (- 0.5 (/ t_0 2.0)) (/ (+ 1.0 t_0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = cos((x * 2.0));
	double t_1 = pow(cos(x), 2.0);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = t_2 / t_1;
	double t_4 = 1.0 + t_3;
	return eps * ((1.0 + (eps * ((eps * ((((t_2 * t_4) / t_1) - ((-0.5 * t_4) + (0.16666666666666666 * t_3))) - 0.16666666666666666)) + ((sin(x) * t_4) / cos(x))))) + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = cos((x * 2.0d0))
    t_1 = cos(x) ** 2.0d0
    t_2 = sin(x) ** 2.0d0
    t_3 = t_2 / t_1
    t_4 = 1.0d0 + t_3
    code = eps * ((1.0d0 + (eps * ((eps * ((((t_2 * t_4) / t_1) - (((-0.5d0) * t_4) + (0.16666666666666666d0 * t_3))) - 0.16666666666666666d0)) + ((sin(x) * t_4) / cos(x))))) + ((0.5d0 - (t_0 / 2.0d0)) / ((1.0d0 + t_0) / 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = t_2 / t_1;
	double t_4 = 1.0 + t_3;
	return eps * ((1.0 + (eps * ((eps * ((((t_2 * t_4) / t_1) - ((-0.5 * t_4) + (0.16666666666666666 * t_3))) - 0.16666666666666666)) + ((Math.sin(x) * t_4) / Math.cos(x))))) + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)));
}

Python

def code(x, eps):
	t_0 = math.cos((x * 2.0))
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = t_2 / t_1
	t_4 = 1.0 + t_3
	return eps * ((1.0 + (eps * ((eps * ((((t_2 * t_4) / t_1) - ((-0.5 * t_4) + (0.16666666666666666 * t_3))) - 0.16666666666666666)) + ((math.sin(x) * t_4) / math.cos(x))))) + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)))

Julia

function code(x, eps)
	t_0 = cos(Float64(x * 2.0))
	t_1 = cos(x) ^ 2.0
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(1.0 + t_3)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(Float64(Float64(t_2 * t_4) / t_1) - Float64(Float64(-0.5 * t_4) + Float64(0.16666666666666666 * t_3))) - 0.16666666666666666)) + Float64(Float64(sin(x) * t_4) / cos(x))))) + Float64(Float64(0.5 - Float64(t_0 / 2.0)) / Float64(Float64(1.0 + t_0) / 2.0))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos((x * 2.0));
	t_1 = cos(x) ^ 2.0;
	t_2 = sin(x) ^ 2.0;
	t_3 = t_2 / t_1;
	t_4 = 1.0 + t_3;
	tmp = eps * ((1.0 + (eps * ((eps * ((((t_2 * t_4) / t_1) - ((-0.5 * t_4) + (0.16666666666666666 * t_3))) - 0.16666666666666666)) + ((sin(x) * t_4) / cos(x))))) + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + t$95$3), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(N[(t$95$2 * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * t$95$4), $MachinePrecision] + N[(0.16666666666666666 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$4), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 100.0%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

5. Applied egg-rr 100.0%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

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

   5. count-2 99.7%

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

   6. *-commutative 99.7%

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

7. Simplified 100.0%

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

   1. unpow2 100.0%

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

   2. cos-mult 100.0%

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

9. Applied egg-rr 100.0%

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

    1. +-commutative 100.0%

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

    2. +-inverses 100.0%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

    3. cos-0 100.0%

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

    4. count-2 100.0%

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

    5. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} - \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}\right) \]
13. Add Preprocessing

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := 1 + t\_2\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} + \varepsilon \cdot \left(\left(\frac{t\_1 \cdot t\_3}{t\_0} + \left(-0.5 \cdot \left(-1 - t\_2\right) - 0.16666666666666666 \cdot t\_1\right)\right) - 0.16666666666666666\right)\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (/ t_1 t_0))
        (t_3 (+ 1.0 t_2)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (-
          (+
           (/ (* t_1 t_3) t_0)
           (- (* -0.5 (- -1.0 t_2)) (* 0.16666666666666666 t_1)))
          0.16666666666666666)))))
     (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = t_1 / t_0;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = cos(x) ** 2.0d0
    t_1 = sin(x) ** 2.0d0
    t_2 = t_1 / t_0
    t_3 = 1.0d0 + t_2
    code = eps * ((1.0d0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + (((-0.5d0) * ((-1.0d0) - t_2)) - (0.16666666666666666d0 * t_1))) - 0.16666666666666666d0))))) + ((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.cos(x), 2.0);
	double t_1 = Math.pow(Math.sin(x), 2.0);
	double t_2 = t_1 / t_0;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * (((Math.sin(x) * t_3) / Math.cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + ((0.5 - (Math.cos((x * 2.0)) / 2.0)) / t_0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	t_1 = math.pow(math.sin(x), 2.0)
	t_2 = t_1 / t_0
	t_3 = 1.0 + t_2
	return eps * ((1.0 + (eps * (((math.sin(x) * t_3) / math.cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + ((0.5 - (math.cos((x * 2.0)) / 2.0)) / t_0))

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	t_3 = Float64(1.0 + t_2)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) + Float64(eps * Float64(Float64(Float64(Float64(t_1 * t_3) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(0.16666666666666666 * t_1))) - 0.16666666666666666))))) + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	t_1 = sin(x) ^ 2.0;
	t_2 = t_1 / t_0;
	t_3 = 1.0 + t_2;
	tmp = eps * ((1.0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 100.0%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

5. Applied egg-rr 100.0%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

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

   5. count-2 99.7%

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

   6. *-commutative 99.7%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - 0.16666666666666666 \cdot {\sin x}^{2}\right)\right) - 0.16666666666666666\right)\right)\right) + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]
10. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t\_0} + \left(1 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{t\_0}\right)}{\cos x} - \varepsilon \cdot -0.3333333333333333\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)))
   (*
    eps
    (+
     (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_0)
     (+
      1.0
      (*
       eps
       (-
        (/ (* (sin x) (+ 1.0 (/ (pow (sin x) 2.0) t_0))) (cos x))
        (* eps -0.3333333333333333))))))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	return eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / t_0) + (1.0 + (eps * (((sin(x) * (1.0 + (pow(sin(x), 2.0) / t_0))) / cos(x)) - (eps * -0.3333333333333333)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = cos(x) ** 2.0d0
    code = eps * (((0.5d0 - (cos((x * 2.0d0)) / 2.0d0)) / t_0) + (1.0d0 + (eps * (((sin(x) * (1.0d0 + ((sin(x) ** 2.0d0) / t_0))) / cos(x)) - (eps * (-0.3333333333333333d0))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.cos(x), 2.0);
	return eps * (((0.5 - (Math.cos((x * 2.0)) / 2.0)) / t_0) + (1.0 + (eps * (((Math.sin(x) * (1.0 + (Math.pow(Math.sin(x), 2.0) / t_0))) / Math.cos(x)) - (eps * -0.3333333333333333)))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	return eps * (((0.5 - (math.cos((x * 2.0)) / 2.0)) / t_0) + (1.0 + (eps * (((math.sin(x) * (1.0 + (math.pow(math.sin(x), 2.0) / t_0))) / math.cos(x)) - (eps * -0.3333333333333333)))))

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	return Float64(eps * Float64(Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_0) + Float64(1.0 + Float64(eps * Float64(Float64(Float64(sin(x) * Float64(1.0 + Float64((sin(x) ^ 2.0) / t_0))) / cos(x)) - Float64(eps * -0.3333333333333333))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	tmp = eps * (((0.5 - (cos((x * 2.0)) / 2.0)) / t_0) + (1.0 + (eps * (((sin(x) * (1.0 + ((sin(x) ^ 2.0) / t_0))) / cos(x)) - (eps * -0.3333333333333333)))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(1.0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 100.0%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

5. Applied egg-rr 100.0%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

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

   5. count-2 99.7%

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

   6. *-commutative 99.7%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 99.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \color{blue}{-0.3333333333333333}\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right) \]

9. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} - \varepsilon \cdot -0.3333333333333333\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{2}\\ t_1 := {\cos x}^{2}\\ \varepsilon \cdot \left(1 + \left(\frac{0.5 - t\_0}{t\_1} + \varepsilon \cdot \frac{\sin x \cdot \left(1 - \frac{t\_0 - 0.5}{t\_1}\right)}{\cos x}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (cos (* x 2.0)) 2.0)) (t_1 (pow (cos x) 2.0)))
   (*
    eps
    (+
     1.0
     (+
      (/ (- 0.5 t_0) t_1)
      (* eps (/ (* (sin x) (- 1.0 (/ (- t_0 0.5) t_1))) (cos x))))))))

C

double code(double x, double eps) {
	double t_0 = cos((x * 2.0)) / 2.0;
	double t_1 = pow(cos(x), 2.0);
	return eps * (1.0 + (((0.5 - t_0) / t_1) + (eps * ((sin(x) * (1.0 - ((t_0 - 0.5) / t_1))) / cos(x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = cos((x * 2.0d0)) / 2.0d0
    t_1 = cos(x) ** 2.0d0
    code = eps * (1.0d0 + (((0.5d0 - t_0) / t_1) + (eps * ((sin(x) * (1.0d0 - ((t_0 - 0.5d0) / t_1))) / cos(x)))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((x * 2.0)) / 2.0;
	double t_1 = Math.pow(Math.cos(x), 2.0);
	return eps * (1.0 + (((0.5 - t_0) / t_1) + (eps * ((Math.sin(x) * (1.0 - ((t_0 - 0.5) / t_1))) / Math.cos(x)))));
}

Python

def code(x, eps):
	t_0 = math.cos((x * 2.0)) / 2.0
	t_1 = math.pow(math.cos(x), 2.0)
	return eps * (1.0 + (((0.5 - t_0) / t_1) + (eps * ((math.sin(x) * (1.0 - ((t_0 - 0.5) / t_1))) / math.cos(x)))))

Julia

function code(x, eps)
	t_0 = Float64(cos(Float64(x * 2.0)) / 2.0)
	t_1 = cos(x) ^ 2.0
	return Float64(eps * Float64(1.0 + Float64(Float64(Float64(0.5 - t_0) / t_1) + Float64(eps * Float64(Float64(sin(x) * Float64(1.0 - Float64(Float64(t_0 - 0.5) / t_1))) / cos(x))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos((x * 2.0)) / 2.0;
	t_1 = cos(x) ^ 2.0;
	tmp = eps * (1.0 + (((0.5 - t_0) / t_1) + (eps * ((sin(x) * (1.0 - ((t_0 - 0.5) / t_1))) / cos(x)))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 + N[(N[(N[(0.5 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[(N[(t$95$0 - 0.5), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

7. Applied egg-rr 99.9%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

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

   5. count-2 99.7%

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

   6. *-commutative 99.7%

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

9. Simplified 99.9%

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

    1. unpow2 99.7%

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

    2. sin-mult 99.7%

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

11. Applied egg-rr 99.9%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}\right)\right)\right) \]
12. Step-by-step derivation

    1. div-sub 99.7%

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

    2. +-inverses 99.7%

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

    3. cos-0 99.7%

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

    4. metadata-eval 99.7%

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

    5. count-2 99.7%

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

    6. *-commutative 99.7%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}} + \varepsilon \cdot \frac{\sin x \cdot \left(1 - \frac{\frac{\cos \left(x \cdot 2\right)}{2} - 0.5}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \]
15. Add Preprocessing

Alternative 6: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 - \frac{\frac{\cos \left(x \cdot 2\right)}{2} - 0.5}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (- 1.0 (/ (- (/ (cos (* x 2.0)) 2.0) 0.5) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 - (((cos((x * 2.0)) / 2.0) - 0.5) / pow(cos(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 - (((cos((x * 2.0d0)) / 2.0d0) - 0.5d0) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 - (((Math.cos((x * 2.0)) / 2.0) - 0.5) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 - (((math.cos((x * 2.0)) / 2.0) - 0.5) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 - Float64(Float64(Float64(cos(Float64(x * 2.0)) / 2.0) - 0.5) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 - (((cos((x * 2.0)) / 2.0) - 0.5) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 - N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. sub-neg 99.7%

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

   2. mul-1-neg 99.7%

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

   3. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

   1. unpow2 99.7%

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

   2. sin-mult 99.7%

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

7. Applied egg-rr 99.7%

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

   1. div-sub 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. metadata-eval 99.7%

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

   5. count-2 99.7%

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

   6. *-commutative 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 - \frac{\frac{\cos \left(x \cdot 2\right)}{2} - 0.5}{{\cos x}^{2}}\right) \]
11. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\varepsilon \cdot x + {\tan x}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (+ (* eps x) (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((eps * x) + pow(tan(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((eps * x) + (tan(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((eps * x) + Math.pow(Math.tan(x), 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + ((eps * x) + math.pow(math.tan(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(eps * x) + (tan(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((eps * x) + (tan(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(eps * x), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 99.7%

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

   1. sub-neg 99.7%

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

   2. add-sqr-sqrt 43.8%

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

   3. sqrt-unprod 99.2%

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

   4. sqr-neg 99.2%

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

   5. sqrt-unprod 99.2%

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

   6. add-sqr-sqrt 99.2%

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

   7. *-commutative 99.2%

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

   8. add-sqr-sqrt 43.8%

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

   9. sqrt-unprod 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot x + {\tan x}^{2}\right)\right) \]
10. Add Preprocessing

Alternative 8: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+ 1.0 (* x (+ eps (* x (+ 1.0 (* (pow x 2.0) 0.6666666666666666))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (pow(x, 2.0) * 0.6666666666666666))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (eps + (x * (1.0d0 + ((x ** 2.0d0) * 0.6666666666666666d0))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (Math.pow(x, 2.0) * 0.6666666666666666))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + (x * (1.0 + (math.pow(x, 2.0) * 0.6666666666666666))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(eps + Float64(x * Float64(1.0 + Float64((x ^ 2.0) * 0.6666666666666666)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (eps + (x * (1.0 + ((x ^ 2.0) * 0.6666666666666666))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(eps + N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 99.7%

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

7. Taylor expanded in x around 0 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + 0.6666666666666666 \cdot {x}^{2}\right)\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right)\right) \]

9. Simplified 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)\right)}\right) \]

10. Final simplification 99.5%

    \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 98.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    x
    (+
     eps
     (*
      x
      (+
       1.0
       (* x (+ (* x 0.6666666666666666) (* eps 1.3333333333333333))))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (eps + (x * (1.0d0 + (x * ((x * 0.6666666666666666d0) + (eps * 1.3333333333333333d0))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(eps + Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.6666666666666666) + Float64(eps * 1.3333333333333333)))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((x * 0.6666666666666666) + (eps * 1.3333333333333333))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(eps + N[(x * N[(1.0 + N[(x * N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(eps * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(0.6666666666666666 \cdot x + 0.8333333333333334 \cdot \varepsilon\right) - -0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate--l+ 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \color{blue}{\left(0.6666666666666666 \cdot x + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)}\right)\right)\right) \]

   2. *-commutative 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\color{blue}{x \cdot 0.6666666666666666} + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)\right) \]

   3. distribute-rgt-out-- 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \color{blue}{\varepsilon \cdot \left(0.8333333333333334 - -0.5\right)}\right)\right)\right)\right) \]

   4. metadata-eval 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot \color{blue}{1.3333333333333333}\right)\right)\right)\right) \]

8. Simplified 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right)\right)\right)}\right) \]

9. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot 1.3333333333333333\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 10: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 11: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. associate--l+ 99.9%

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

   2. associate-/l* 99.9%

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

   3. mul-1-neg 99.9%

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

   4. mul-1-neg 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around 0 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

   1. distribute-rgt-in 99.4%

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

   2. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 12: 98.0% 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 61.4%

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

3. Taylor expanded in x around 0 99.3%

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

4. Taylor expanded in eps around 0 99.3%

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

5. Final simplification 99.3%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 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 2024084 
(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
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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