2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 38.7s

Alternatives: 8

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return eps * ((log(pow(exp(eps), fma(-eps, (0.16666666666666666 + fma(-1.0, pow(((sin(x) * hypot(1.0, tan(x))) / cos(x)), 2.0), fma(0.16666666666666666, t_0, (-0.5 + (t_0 * -0.5))))), (sin(x) * (((pow(sin(x), 2.0) * pow(cos(x), -2.0)) + 1.0) / cos(x)))))) + 1.0) + (((1.0 - cos((x * 2.0))) / 2.0) / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(eps * Float64(Float64(log((exp(eps) ^ fma(Float64(-eps), Float64(0.16666666666666666 + fma(-1.0, (Float64(Float64(sin(x) * hypot(1.0, tan(x))) / cos(x)) ^ 2.0), fma(0.16666666666666666, t_0, Float64(-0.5 + Float64(t_0 * -0.5))))), Float64(sin(x) * Float64(Float64(Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0)) + 1.0) / cos(x)))))) + 1.0) + Float64(Float64(Float64(1.0 - cos(Float64(x * 2.0))) / 2.0) / (cos(x) ^ 2.0))))
end

Wolfram

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in eps around 0 99.5%

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

5. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

   2. sin-mult 99.5%

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

7. Applied egg-rr 99.5%

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

   1. +-inverses 99.5%

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

   2. cos-0 99.5%

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

   3. count-2 99.5%

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

   4. *-commutative 99.5%

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

9. Simplified 99.5%

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

11. Applied egg-rr 99.5%

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

    1. *-lft-identity 99.5%

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

    2. associate-*r/ 99.5%

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

    3. *-commutative 99.5%

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

    4. distribute-rgt-in 99.5%

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

    5. metadata-eval 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

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))
            (* t_0 -0.3333333333333333)))))
        (+ (/ (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)) - (t_0 * -0.3333333333333333))))) + ((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)) - (t_0 * (-0.3333333333333333d0)))))) + ((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)) - (t_0 * -0.3333333333333333))))) + ((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)) - (t_0 * -0.3333333333333333))))) + ((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(t_0 * -0.3333333333333333))))) + 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)) - (t_0 * -0.3333333333333333))))) + ((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[(t$95$0 * -0.3333333333333333), $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 59.6%

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

   1. tan-sum 60.0%

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

   2. div-inv 60.0%

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

   3. fma-neg 60.0%

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

4. Applied egg-rr 60.0%

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

   1. fma-neg 60.0%

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

   2. associate-*r/ 60.0%

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

   3. *-rgt-identity 60.0%

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

6. Simplified 60.0%

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

7. Taylor expanded in eps around 0 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \tan x + {\tan x}^{3}\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (+
    (* eps (fma eps 0.3333333333333333 (+ (tan x) (pow (tan x) 3.0))))
    1.0))))

C

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

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(Float64(eps * fma(eps, 0.3333333333333333, Float64(tan(x) + (tan(x) ^ 3.0)))) + 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(eps * 0.3333333333333333 + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 99.3%

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

6. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. distribute-rgt-neg-in 99.3%

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

   3. metadata-eval 99.3%

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

   4. distribute-rgt-in 99.3%

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

   5. fma-define 99.3%

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

   6. neg-mul-1 99.3%

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

   7. neg-sub0 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{\left(0 - \mathsf{fma}\left(\varepsilon, -0.3333333333333333, \frac{1 + {\tan x}^{2}}{1} \cdot \tan x\right)\right)} + \mathsf{fma}\left(\frac{1 + {\tan x}^{2}}{1} \cdot \tan x, 1, \left(\frac{1 + {\tan x}^{2}}{1} \cdot \tan x\right) \cdot 1\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

   8. fma-undefine 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\left(0 - \mathsf{fma}\left(\varepsilon, -0.3333333333333333, \frac{1 + {\tan x}^{2}}{1} \cdot \tan x\right)\right) + \color{blue}{\left(\left(\frac{1 + {\tan x}^{2}}{1} \cdot \tan x\right) \cdot 1 + \left(\frac{1 + {\tan x}^{2}}{1} \cdot \tan x\right) \cdot 1\right)}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\varepsilon \cdot -0.3333333333333333 + \tan x \cdot \left({\tan x}^{2} + 1\right)\right) + 1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 99.3%

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

   1. cancel-sign-sub-inv 99.3%

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

   2. add-sqr-sqrt 99.3%

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

   3. sqrt-unprod 99.3%

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

   4. mul-1-neg 99.3%

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

   5. mul-1-neg 99.3%

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

   6. sqr-neg 99.3%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 99.2%

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

   9. metadata-eval 99.2%

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

   10. *-un-lft-identity 99.2%

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

   11. *-commutative 99.2%

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

   12. *-un-lft-identity 99.2%

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

   13. times-frac 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in eps around 0 98.8%

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

   1. sub-neg 98.8%

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

   2. mul-1-neg 98.8%

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

   3. remove-double-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in x around 0 97.2%

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

4. Final simplification 97.2%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. Add Preprocessing

Alternative 7: 98.1% 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 59.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. tan-quot 97.2%

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

   2. *-un-lft-identity 97.2%

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

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. Final simplification 97.2%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 8: 98.1% 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 59.6%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Final simplification 97.2%

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

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024067 
(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)))