2tan (problem 3.3.2)

Percentage Accurate: 85.9% → 98.8%

Time: 22.9s

Alternatives: 13

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: 85.9% 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: 98.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := t\_2 + 1\\ t_4 := \frac{t\_1 \cdot t\_3}{t\_0}\\ t_5 := -1 - t\_2\\ t_6 := \left(-0.5 \cdot t\_5 - t\_2 \cdot 0.16666666666666666\right) - 0.16666666666666666\\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{\sin x \cdot t\_3}{\cos x} + \varepsilon \cdot \left(\left(t\_4 + \varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot t\_5}{\cos x} + \frac{\sin x \cdot \left(t\_4 + t\_6\right)}{\cos x}\right)\right) + t\_6\right), \frac{{\log \left(e^{\sin x}\right)}^{2}}{t\_0}\right) + 1\right) \end{array} \end{array} \]

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 (+ t_2 1.0))
        (t_4 (/ (* t_1 t_3) t_0))
        (t_5 (- -1.0 t_2))
        (t_6
         (- (- (* -0.5 t_5) (* t_2 0.16666666666666666)) 0.16666666666666666)))
   (*
    eps
    (+
     (fma
      eps
      (+
       (/ (* (sin x) t_3) (cos x))
       (*
        eps
        (+
         (+
          t_4
          (*
           eps
           (+
            (* -0.3333333333333333 (/ (* (sin x) t_5) (cos x)))
            (/ (* (sin x) (+ t_4 t_6)) (cos x)))))
         t_6)))
      (/ (pow (log (exp (sin x))) 2.0) t_0))
     1.0))))

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 = t_2 + 1.0;
	double t_4 = (t_1 * t_3) / t_0;
	double t_5 = -1.0 - t_2;
	double t_6 = ((-0.5 * t_5) - (t_2 * 0.16666666666666666)) - 0.16666666666666666;
	return eps * (fma(eps, (((sin(x) * t_3) / cos(x)) + (eps * ((t_4 + (eps * ((-0.3333333333333333 * ((sin(x) * t_5) / cos(x))) + ((sin(x) * (t_4 + t_6)) / cos(x))))) + t_6))), (pow(log(exp(sin(x))), 2.0) / t_0)) + 1.0);
}

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(t_2 + 1.0)
	t_4 = Float64(Float64(t_1 * t_3) / t_0)
	t_5 = Float64(-1.0 - t_2)
	t_6 = Float64(Float64(Float64(-0.5 * t_5) - Float64(t_2 * 0.16666666666666666)) - 0.16666666666666666)
	return Float64(eps * Float64(fma(eps, Float64(Float64(Float64(sin(x) * t_3) / cos(x)) + Float64(eps * Float64(Float64(t_4 + Float64(eps * Float64(Float64(-0.3333333333333333 * Float64(Float64(sin(x) * t_5) / cos(x))) + Float64(Float64(sin(x) * Float64(t_4 + t_6)) / cos(x))))) + t_6))), Float64((log(exp(sin(x))) ^ 2.0) / t_0)) + 1.0))
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[(t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 - t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(-0.5 * t$95$5), $MachinePrecision] - N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(t$95$4 + N[(eps * N[(N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] * t$95$5), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 + t$95$6), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[N[Exp[N[Sin[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\color{blue}{\log \left(e^{\sin x}\right)}}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\color{blue}{\log \left(e^{\sin x}\right)}}^{2}}{{\cos x}^{2}}\right)\right) \]

8. Taylor expanded in eps around 0 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\left(-1 \cdot \left(\varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) - \left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}}, \frac{{\log \left(e^{\sin x}\right)}^{2}}{{\cos x}^{2}}\right)\right) \]

9. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} + \varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{{\cos x}^{2}} + \varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{{\sin x}^{2} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{{\cos x}^{2}} + \left(\left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) - 0.16666666666666666\right)\right)}{\cos x}\right)\right) + \left(\left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) - 0.16666666666666666\right)\right), \frac{{\log \left(e^{\sin x}\right)}^{2}}{{\cos x}^{2}}\right) + 1\right) \]
10. Add Preprocessing

Alternative 2: 98.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ t_4 := -1 - t\_2\\ t_5 := \left(-0.5 \cdot t\_4 - t\_2 \cdot 0.16666666666666666\right) - 0.16666666666666666\\ t_6 := \frac{t\_0 \cdot t\_3}{t\_1}\\ \varepsilon \cdot \left(\left(t\_2 + \varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} + \varepsilon \cdot \left(\left(t\_6 + \varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot t\_4}{\cos x} + \frac{\sin x \cdot \left(t\_6 + t\_5\right)}{\cos x}\right)\right) + t\_5\right)\right)\right) + 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0))
        (t_4 (- -1.0 t_2))
        (t_5
         (- (- (* -0.5 t_4) (* t_2 0.16666666666666666)) 0.16666666666666666))
        (t_6 (/ (* t_0 t_3) t_1)))
   (*
    eps
    (+
     (+
      t_2
      (*
       eps
       (+
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (+
          (+
           t_6
           (*
            eps
            (+
             (* -0.3333333333333333 (/ (* (sin x) t_4) (cos x)))
             (/ (* (sin x) (+ t_6 t_5)) (cos x)))))
          t_5)))))
     1.0))))

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 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	double t_4 = -1.0 - t_2;
	double t_5 = ((-0.5 * t_4) - (t_2 * 0.16666666666666666)) - 0.16666666666666666;
	double t_6 = (t_0 * t_3) / t_1;
	return eps * ((t_2 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((t_6 + (eps * ((-0.3333333333333333 * ((sin(x) * t_4) / cos(x))) + ((sin(x) * (t_6 + t_5)) / cos(x))))) + t_5))))) + 1.0);
}

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
    real(8) :: t_5
    real(8) :: t_6
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    t_4 = (-1.0d0) - t_2
    t_5 = (((-0.5d0) * t_4) - (t_2 * 0.16666666666666666d0)) - 0.16666666666666666d0
    t_6 = (t_0 * t_3) / t_1
    code = eps * ((t_2 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((t_6 + (eps * (((-0.3333333333333333d0) * ((sin(x) * t_4) / cos(x))) + ((sin(x) * (t_6 + t_5)) / cos(x))))) + t_5))))) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	double t_4 = -1.0 - t_2;
	double t_5 = ((-0.5 * t_4) - (t_2 * 0.16666666666666666)) - 0.16666666666666666;
	double t_6 = (t_0 * t_3) / t_1;
	return eps * ((t_2 + (eps * (((Math.sin(x) * t_3) / Math.cos(x)) + (eps * ((t_6 + (eps * ((-0.3333333333333333 * ((Math.sin(x) * t_4) / Math.cos(x))) + ((Math.sin(x) * (t_6 + t_5)) / Math.cos(x))))) + t_5))))) + 1.0);
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = t_2 + 1.0
	t_4 = -1.0 - t_2
	t_5 = ((-0.5 * t_4) - (t_2 * 0.16666666666666666)) - 0.16666666666666666
	t_6 = (t_0 * t_3) / t_1
	return eps * ((t_2 + (eps * (((math.sin(x) * t_3) / math.cos(x)) + (eps * ((t_6 + (eps * ((-0.3333333333333333 * ((math.sin(x) * t_4) / math.cos(x))) + ((math.sin(x) * (t_6 + t_5)) / math.cos(x))))) + t_5))))) + 1.0)

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	t_4 = Float64(-1.0 - t_2)
	t_5 = Float64(Float64(Float64(-0.5 * t_4) - Float64(t_2 * 0.16666666666666666)) - 0.16666666666666666)
	t_6 = Float64(Float64(t_0 * t_3) / t_1)
	return Float64(eps * Float64(Float64(t_2 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) + Float64(eps * Float64(Float64(t_6 + Float64(eps * Float64(Float64(-0.3333333333333333 * Float64(Float64(sin(x) * t_4) / cos(x))) + Float64(Float64(sin(x) * Float64(t_6 + t_5)) / cos(x))))) + t_5))))) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	t_4 = -1.0 - t_2;
	t_5 = ((-0.5 * t_4) - (t_2 * 0.16666666666666666)) - 0.16666666666666666;
	t_6 = (t_0 * t_3) / t_1;
	tmp = eps * ((t_2 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((t_6 + (eps * ((-0.3333333333333333 * ((sin(x) * t_4) / cos(x))) + ((sin(x) * (t_6 + t_5)) / cos(x))))) + t_5))))) + 1.0);
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), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 - t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(-0.5 * t$95$4), $MachinePrecision] - N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(eps * N[(N[(t$95$2 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(t$95$6 + N[(eps * N[(N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] * t$95$4), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$6 + t$95$5), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.5%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(-1 \cdot \left(\varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) - \left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{\cos x} + \varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{{\cos x}^{2}} + \varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{{\sin x}^{2} \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)}{{\cos x}^{2}} + \left(\left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) - 0.16666666666666666\right)\right)}{\cos x}\right)\right) + \left(\left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) - 0.16666666666666666\right)\right)\right)\right) + 1\right) \]
8. Add Preprocessing

Alternative 3: 98.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_4 := -0.3333333333333333 \cdot t\_3\\ \varepsilon \cdot \left(t\_3 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(t\_3 + 0.3333333333333333\right)}{\cos x} - \left(-0.3333333333333333 \cdot t\_1 + \left(-0.3333333333333333 \cdot t\_2 + \frac{\sin x \cdot \left(t\_4 - t\_0\right)}{\cos x}\right)\right)\right)\right) + \left(t\_3 + \left(t\_0 - t\_4\right)\right)\right) + \left(t\_1 + t\_2\right)\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_4 (* -0.3333333333333333 t_3)))
   (*
    eps
    (+
     t_3
     (+
      (*
       eps
       (+
        (*
         eps
         (+
          (+
           0.3333333333333333
           (*
            eps
            (-
             (/ (* (sin x) (+ t_3 0.3333333333333333)) (cos x))
             (+
              (* -0.3333333333333333 t_1)
              (+
               (* -0.3333333333333333 t_2)
               (/ (* (sin x) (- t_4 t_0)) (cos x)))))))
          (+ t_3 (- t_0 t_4))))
        (+ t_1 t_2)))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 4.0) / pow(cos(x), 4.0);
	double t_1 = sin(x) / cos(x);
	double t_2 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_4 = -0.3333333333333333 * t_3;
	return eps * (t_3 + ((eps * ((eps * ((0.3333333333333333 + (eps * (((sin(x) * (t_3 + 0.3333333333333333)) / cos(x)) - ((-0.3333333333333333 * t_1) + ((-0.3333333333333333 * t_2) + ((sin(x) * (t_4 - t_0)) / cos(x))))))) + (t_3 + (t_0 - t_4)))) + (t_1 + t_2))) + 1.0));
}

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 = (sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)
    t_1 = sin(x) / cos(x)
    t_2 = (sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)
    t_3 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    t_4 = (-0.3333333333333333d0) * t_3
    code = eps * (t_3 + ((eps * ((eps * ((0.3333333333333333d0 + (eps * (((sin(x) * (t_3 + 0.3333333333333333d0)) / cos(x)) - (((-0.3333333333333333d0) * t_1) + (((-0.3333333333333333d0) * t_2) + ((sin(x) * (t_4 - t_0)) / cos(x))))))) + (t_3 + (t_0 - t_4)))) + (t_1 + t_2))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0);
	double t_1 = Math.sin(x) / Math.cos(x);
	double t_2 = Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0);
	double t_3 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	double t_4 = -0.3333333333333333 * t_3;
	return eps * (t_3 + ((eps * ((eps * ((0.3333333333333333 + (eps * (((Math.sin(x) * (t_3 + 0.3333333333333333)) / Math.cos(x)) - ((-0.3333333333333333 * t_1) + ((-0.3333333333333333 * t_2) + ((Math.sin(x) * (t_4 - t_0)) / Math.cos(x))))))) + (t_3 + (t_0 - t_4)))) + (t_1 + t_2))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)
	t_1 = math.sin(x) / math.cos(x)
	t_2 = math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)
	t_3 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	t_4 = -0.3333333333333333 * t_3
	return eps * (t_3 + ((eps * ((eps * ((0.3333333333333333 + (eps * (((math.sin(x) * (t_3 + 0.3333333333333333)) / math.cos(x)) - ((-0.3333333333333333 * t_1) + ((-0.3333333333333333 * t_2) + ((math.sin(x) * (t_4 - t_0)) / math.cos(x))))))) + (t_3 + (t_0 - t_4)))) + (t_1 + t_2))) + 1.0))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))
	t_1 = Float64(sin(x) / cos(x))
	t_2 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_4 = Float64(-0.3333333333333333 * t_3)
	return Float64(eps * Float64(t_3 + Float64(Float64(eps * Float64(Float64(eps * Float64(Float64(0.3333333333333333 + Float64(eps * Float64(Float64(Float64(sin(x) * Float64(t_3 + 0.3333333333333333)) / cos(x)) - Float64(Float64(-0.3333333333333333 * t_1) + Float64(Float64(-0.3333333333333333 * t_2) + Float64(Float64(sin(x) * Float64(t_4 - t_0)) / cos(x))))))) + Float64(t_3 + Float64(t_0 - t_4)))) + Float64(t_1 + t_2))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 4.0) / (cos(x) ^ 4.0);
	t_1 = sin(x) / cos(x);
	t_2 = (sin(x) ^ 3.0) / (cos(x) ^ 3.0);
	t_3 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	t_4 = -0.3333333333333333 * t_3;
	tmp = eps * (t_3 + ((eps * ((eps * ((0.3333333333333333 + (eps * (((sin(x) * (t_3 + 0.3333333333333333)) / cos(x)) - ((-0.3333333333333333 * t_1) + ((-0.3333333333333333 * t_2) + ((sin(x) * (t_4 - t_0)) / cos(x))))))) + (t_3 + (t_0 - t_4)))) + (t_1 + t_2))) + 1.0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.3333333333333333 * t$95$3), $MachinePrecision]}, N[(eps * N[(t$95$3 + N[(N[(eps * N[(N[(eps * N[(N[(0.3333333333333333 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$3 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.3333333333333333 * t$95$1), $MachinePrecision] + N[(N[(-0.3333333333333333 * t$95$2), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$0 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 85.9%

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

   1. tan-sum 86.2%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 86.1%

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

   3. fmm-def 86.1%

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

4. Applied egg-rr 86.1%

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

5. Taylor expanded in eps around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 + -1 \cdot \left(\varepsilon \cdot \left(-1 \cdot \frac{\sin x \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right)\right)\right) - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 + \varepsilon \cdot \left(\frac{\sin x \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 0.3333333333333333\right)}{\cos x} - \left(-0.3333333333333333 \cdot \frac{\sin x}{\cos x} + \left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)}{\cos x}\right)\right)\right)\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + 1\right)\right) \]
7. Add Preprocessing

Alternative 4: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := t\_2 + 1\\ t_4 := t\_3 \cdot \frac{\sin x}{\cos x}\\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{-0.3333333333333333}{\cos x} + -0.3333333333333333 \cdot t\_4, -0.16666666666666666\right) + \left(t\_0 \cdot \frac{t\_3}{t\_1} - \mathsf{fma}\left(-0.5, t\_3, \frac{t\_0 \cdot 0.16666666666666666}{t\_1}\right)\right), t\_4\right), t\_2\right) + 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ t_2 1.0))
        (t_4 (* t_3 (/ (sin x) (cos x)))))
   (*
    eps
    (+
     (fma
      eps
      (fma
       eps
       (+
        (fma
         (- eps)
         (+
          (* (sin x) (/ -0.3333333333333333 (cos x)))
          (* -0.3333333333333333 t_4))
         -0.16666666666666666)
        (-
         (* t_0 (/ t_3 t_1))
         (fma -0.5 t_3 (/ (* t_0 0.16666666666666666) t_1))))
       t_4)
      t_2)
     1.0))))

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 = t_0 / t_1;
	double t_3 = t_2 + 1.0;
	double t_4 = t_3 * (sin(x) / cos(x));
	return eps * (fma(eps, fma(eps, (fma(-eps, ((sin(x) * (-0.3333333333333333 / cos(x))) + (-0.3333333333333333 * t_4)), -0.16666666666666666) + ((t_0 * (t_3 / t_1)) - fma(-0.5, t_3, ((t_0 * 0.16666666666666666) / t_1)))), t_4), t_2) + 1.0);
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	t_4 = Float64(t_3 * Float64(sin(x) / cos(x)))
	return Float64(eps * Float64(fma(eps, fma(eps, Float64(fma(Float64(-eps), Float64(Float64(sin(x) * Float64(-0.3333333333333333 / cos(x))) + Float64(-0.3333333333333333 * t_4)), -0.16666666666666666) + Float64(Float64(t_0 * Float64(t_3 / t_1)) - fma(-0.5, t_3, Float64(Float64(t_0 * 0.16666666666666666) / t_1)))), t_4), t_2) + 1.0))
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), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(eps * N[(N[((-eps) * N[(N[(N[Sin[x], $MachinePrecision] * N[(-0.3333333333333333 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * t$95$4), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + N[(N[(t$95$0 * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * t$95$3 + N[(N[(t$95$0 * 0.16666666666666666), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{\color{blue}{-0.3333333333333333}}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{-0.3333333333333333}{\cos x} + -0.3333333333333333 \cdot \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \cdot \frac{\sin x}{\cos x}\right), -0.16666666666666666\right) + \left({\sin x}^{2} \cdot \frac{\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1}{{\cos x}^{2}} - \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1, \frac{{\sin x}^{2} \cdot 0.16666666666666666}{{\cos x}^{2}}\right)\right), \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right) \]
8. Add Preprocessing

Alternative 5: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

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

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(t_2 + 1.0)
	return Float64(eps * Float64(fma(eps, Float64(Float64(Float64(sin(x) * t_3) / cos(x)) + Float64(eps * Float64(Float64(Float64(t_0 * t_3) / t_1) + Float64(Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(t_2 * 0.16666666666666666)) - 0.16666666666666666)))), Float64((log(exp(sin(x))) ^ 2.0) / t_1)) + 1.0))
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), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[N[Exp[N[Sin[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
6. Step-by-step derivation

   1. add-log-exp 99.5%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\color{blue}{\log \left(e^{\sin x}\right)}}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\color{blue}{\log \left(e^{\sin x}\right)}}^{2}}{{\cos x}^{2}}\right)\right) \]

8. Taylor expanded in eps around 0 99.5%

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

9. Final simplification 99.5%

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

Alternative 6: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

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

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 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = t_2 + 1.0d0
    code = eps * ((t_2 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * (((t_0 * t_3) / t_1) + ((((-0.5d0) * ((-1.0d0) - t_2)) - (t_2 * 0.16666666666666666d0)) - 0.16666666666666666d0)))))) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = t_2 + 1.0;
	tmp = eps * ((t_2 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * (((t_0 * t_3) / t_1) + (((-0.5 * (-1.0 - t_2)) - (t_2 * 0.16666666666666666)) - 0.16666666666666666)))))) + 1.0);
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), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, N[(eps * N[(N[(t$95$2 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 7: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(t\_0 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 + \left(t\_0 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - -0.3333333333333333 \cdot t\_0\right)\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + 1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     t_0
     (+
      (*
       eps
       (+
        (*
         eps
         (+
          0.3333333333333333
          (+
           t_0
           (-
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
            (* -0.3333333333333333 t_0)))))
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
      1.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 + (t_0 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (-0.3333333333333333 * t_0))))) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * (t_0 + ((eps * ((eps * (0.3333333333333333d0 + (t_0 + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * t_0))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 + (t_0 + ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (-0.3333333333333333 * t_0))))) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))))) + 1.0));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (t_0 + ((eps * ((eps * (0.3333333333333333 + (t_0 + ((math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (-0.3333333333333333 * t_0))))) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))) + 1.0))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(t_0 + Float64(Float64(eps * Float64(Float64(eps * Float64(0.3333333333333333 + Float64(t_0 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(-0.3333333333333333 * t_0))))) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (t_0 + ((eps * ((eps * (0.3333333333333333 + (t_0 + (((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - (-0.3333333333333333 * t_0))))) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))) + 1.0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 + N[(N[(eps * N[(N[(eps * N[(0.3333333333333333 + N[(t$95$0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 85.9%

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

   1. tan-sum 86.2%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 86.1%

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

   3. fmm-def 86.1%

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

4. Applied egg-rr 86.1%

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

5. Taylor expanded in eps around 0 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + 1\right)\right) \]
7. Add Preprocessing

Alternative 8: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.3333333333333333 + x \cdot \left(\left(0.6666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot 1.3333333333333333 + x \cdot \left(1.3333333333333333 + {\varepsilon}^{2} \cdot 1.8888888888888888\right)\right)\right) + 1\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (fma
    eps
    (+
     (* eps 0.3333333333333333)
     (*
      x
      (+
       (+
        (* 0.6666666666666666 (pow eps 2.0))
        (*
         x
         (+
          (* eps 1.3333333333333333)
          (* x (+ 1.3333333333333333 (* (pow eps 2.0) 1.8888888888888888))))))
       1.0)))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
   1.0)))

C

double code(double x, double eps) {
	return eps * (fma(eps, ((eps * 0.3333333333333333) + (x * (((0.6666666666666666 * pow(eps, 2.0)) + (x * ((eps * 1.3333333333333333) + (x * (1.3333333333333333 + (pow(eps, 2.0) * 1.8888888888888888)))))) + 1.0))), (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * Float64(fma(eps, Float64(Float64(eps * 0.3333333333333333) + Float64(x * Float64(Float64(Float64(0.6666666666666666 * (eps ^ 2.0)) + Float64(x * Float64(Float64(eps * 1.3333333333333333) + Float64(x * Float64(1.3333333333333333 + Float64((eps ^ 2.0) * 1.8888888888888888)))))) + 1.0))), Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(x * N[(N[(N[(0.6666666666666666 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(eps * 1.3333333333333333), $MachinePrecision] + N[(x * N[(1.3333333333333333 + N[(N[Power[eps, 2.0], $MachinePrecision] * 1.8888888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Simplified 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333 \cdot \varepsilon + x \cdot \left(1 + \left(0.6666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(1.3333333333333333 \cdot \varepsilon + x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot {\varepsilon}^{2}\right)\right)\right)\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

7. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.3333333333333333 + x \cdot \left(\left(0.6666666666666666 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot 1.3333333333333333 + x \cdot \left(1.3333333333333333 + {\varepsilon}^{2} \cdot 1.8888888888888888\right)\right)\right) + 1\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 1\right) \]
8. Add Preprocessing

Alternative 9: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* eps (* 2.0 (/ (pow (sin x) 2.0) (+ (cos (* x 2.0)) 1.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. mul-1-neg 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

   1. unpow2 98.7%

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

   2. cos-mult 98.7%

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

7. Applied egg-rr 98.7%

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

   1. +-commutative 98.7%

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

   2. +-inverses 98.7%

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

   3. cos-0 98.7%

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

   4. count-2 98.7%

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

   5. *-commutative 98.7%

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

9. Simplified 98.7%

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

    1. distribute-rgt-in 98.7%

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

    2. *-un-lft-identity 98.7%

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

    3. associate-/r/ 98.7%

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 10: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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 * ((((1.0d0 - t_0) / 2.0d0) / ((t_0 + 1.0d0) / 2.0d0)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. mul-1-neg 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

   1. unpow2 98.7%

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

   2. cos-mult 98.7%

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

7. Applied egg-rr 98.7%

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

   1. +-commutative 98.7%

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

   2. +-inverses 98.7%

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

   3. cos-0 98.7%

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

   4. count-2 98.7%

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

   5. *-commutative 98.7%

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

9. Simplified 98.7%

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

    1. unpow2 98.7%

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

    2. sin-mult 98.7%

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

11. Applied egg-rr 98.7%

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

    1. +-inverses 98.7%

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

    2. cos-0 98.7%

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

    3. count-2 98.7%

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

    4. *-commutative 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 11: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in x around 0 98.5%

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

   1. tan-quot 98.5%

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

   2. *-un-lft-identity 98.5%

      \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]

5. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 98.5%

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

7. Simplified 98.5%

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

Alternative 12: 97.8% 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 85.9%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. mul-1-neg 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 98.4%

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

   1. unpow2 98.4%

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

8. Applied egg-rr 98.4%

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

Alternative 13: 97.7% 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 85.9%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. mul-1-neg 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 98.2%

   \[\leadsto \varepsilon \cdot \color{blue}{1} \]

7. Taylor expanded in eps around 0 98.2%

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

Developer Target 1: 99.8% 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]

Developer Target 2: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024169 
(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)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

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