2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.6%

Time: 24.6s

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: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

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

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

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

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

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.0× speedup

Math

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

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
     (fma
      eps
      (fma
       eps
       (+
        (-
         (/ (* t_1 t_3) t_0)
         (*
          eps
          (+
           (* -0.3333333333333333 (/ (* (sin x) t_3) (cos x)))
           (/
            (*
             (sin x)
             (-
              -0.3333333333333333
              (pow (* (sin x) (/ (hypot 1.0 (tan x)) (cos x))) 2.0)))
            (cos x)))))
        (-
         (- (* -0.5 (- -1.0 t_2)) (* t_2 0.16666666666666666))
         0.16666666666666666))
       (* t_3 (/ (sin x) (cos x))))
      t_2)))))

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 + fma(eps, fma(eps, ((((t_1 * t_3) / t_0) - (eps * ((-0.3333333333333333 * ((sin(x) * t_3) / cos(x))) + ((sin(x) * (-0.3333333333333333 - pow((sin(x) * (hypot(1.0, tan(x)) / cos(x))), 2.0))) / cos(x))))) + (((-0.5 * (-1.0 - t_2)) - (t_2 * 0.16666666666666666)) - 0.16666666666666666)), (t_3 * (sin(x) / cos(x)))), t_2));
}

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(1.0 + fma(eps, fma(eps, Float64(Float64(Float64(Float64(t_1 * t_3) / t_0) - Float64(eps * Float64(Float64(-0.3333333333333333 * Float64(Float64(sin(x) * t_3) / cos(x))) + Float64(Float64(sin(x) * Float64(-0.3333333333333333 - (Float64(sin(x) * Float64(hypot(1.0, tan(x)) / cos(x))) ^ 2.0))) / cos(x))))) + Float64(Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(t_2 * 0.16666666666666666)) - 0.16666666666666666)), Float64(t_3 * Float64(sin(x) / cos(x)))), t_2)))
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[(1.0 + N[(eps * N[(eps * N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(eps * N[(N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(-0.3333333333333333 - N[Power[N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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] + N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around 0 99.7%

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

7. Taylor expanded in x around 0 99.7%

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

   1. pow1 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow1 99.7%

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

    2. associate-/l* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, Float64(fma(eps, Float64(Float64(t_0 + 0.3333333333333333) - fma(-0.3333333333333333, t_0, Float64((sin(x) ^ 4.0) / Float64(-(cos(x) ^ 4.0))))), Float64(sin(x) / cos(x))) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))), t_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[(1.0 + N[(eps * N[(N[(eps * N[(N[(t$95$0 + 0.3333333333333333), $MachinePrecision] - N[(-0.3333333333333333 * t$95$0 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / (-N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.4%

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

   1. tan-sum 64.8%

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

   2. div-inv 64.7%

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

   3. fma-neg 64.7%

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

4. Applied egg-rr 64.7%

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

5. Taylor expanded in eps around 0 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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 + (1.0d0 + (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)))))))
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 + (1.0 + (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)))))));
}

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 + (1.0 + (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)))))))

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (t_0 + (1.0 + (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)))))));
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[(1.0 + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.4%

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

   1. tan-sum 64.8%

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

   2. div-inv 64.7%

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

   3. fma-neg 64.7%

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

4. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*l/ 64.8%

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

   4. *-lft-identity 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in eps around 0 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + fma(eps, fma(eps, 0.3333333333333333, Float64(Float64(1.0 + t_0) * Float64(sin(x) / cos(x)))), t_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[(1.0 + N[(eps * N[(eps * 0.3333333333333333 + N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
7. Add Preprocessing

Alternative 6: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
   (* eps (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + Float64(eps * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. tan-sum 64.8%

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

   2. div-inv 64.7%

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

   3. fma-neg 64.7%

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

4. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
5. Step-by-step derivation

   1. fma-neg 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-*l/ 64.8%

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

   4. *-lft-identity 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in eps around 0 99.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + -1 \cdot \left(\varepsilon \cdot \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-+l+ 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 7: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(sin(x), 2.0) / log(exp(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 + ((sin(x) ** 2.0d0) / log(exp((cos(x) ** 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. add-log-exp 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\color{blue}{\log \left(e^{{\cos x}^{2}}\right)}}\right) \]
8. Add Preprocessing

Alternative 8: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(sin(x), 2.0) / 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 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

Alternative 9: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 97.9%

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

   1. *-commutative 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 11: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 97.9%

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

   1. *-commutative 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 12: 97.8% 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 64.4%

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

3. Taylor expanded in x around 0 97.2%

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

   1. *-un-lft-identity 97.2%

      \[\leadsto \color{blue}{1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. quot-tan 97.2%

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

5. Applied egg-rr 97.2%

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

   1. *-lft-identity 97.2%

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

7. Simplified 97.2%

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

Alternative 13: 97.8% 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 64.4%

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

3. Taylor expanded in x around 0 97.2%

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

4. Taylor expanded in eps around 0 97.2%

   \[\leadsto \color{blue}{\varepsilon} \]
5. 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 2024102 
(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)))