2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.7%

Time: 28.7s

Alternatives: 12

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \frac{\varepsilon \cdot \left(\cos x + \left(t\_0 + {\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot t\_0\right)\right) + 0.3333333333333333 \cdot t\_0\right)\right)\right)}{\cos x \cdot \left(1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (+
      (cos x)
      (+
       t_0
       (*
        (pow eps 2.0)
        (+
         (fma
          0.3333333333333333
          (cos x)
          (*
           (pow eps 2.0)
           (- (* (cos x) 0.13333333333333333) (* -0.13333333333333333 t_0))))
         (* 0.3333333333333333 t_0))))))
    (* (cos x) (+ 1.0 (+ 1.0 (- -1.0 (* (tan x) (tan eps)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / cos(x);
	return (eps * (cos(x) + (t_0 + (pow(eps, 2.0) * (fma(0.3333333333333333, cos(x), (pow(eps, 2.0) * ((cos(x) * 0.13333333333333333) - (-0.13333333333333333 * t_0)))) + (0.3333333333333333 * t_0)))))) / (cos(x) * (1.0 + (1.0 + (-1.0 - (tan(x) * tan(eps))))));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / cos(x))
	return Float64(Float64(eps * Float64(cos(x) + Float64(t_0 + Float64((eps ^ 2.0) * Float64(fma(0.3333333333333333, cos(x), Float64((eps ^ 2.0) * Float64(Float64(cos(x) * 0.13333333333333333) - Float64(-0.13333333333333333 * t_0)))) + Float64(0.3333333333333333 * t_0)))))) / Float64(cos(x) * Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(tan(x) * tan(eps)))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(0.3333333333333333 * N[Cos[x], $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - N[(-0.13333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 99.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\cos x + \left({\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan \varepsilon\right)\right)}\right) \cdot \cos x} \]

   2. expm1-undefine 99.7%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan x \cdot \tan \varepsilon\right)} - 1\right)}\right) \cdot \cos x} \]

   3. log1p-undefine 99.7%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \left(e^{\color{blue}{\log \left(1 + \tan x \cdot \tan \varepsilon\right)}} - 1\right)\right) \cdot \cos x} \]

   4. add-exp-log 99.7%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \left(\color{blue}{\left(1 + \tan x \cdot \tan \varepsilon\right)} - 1\right)\right) \cdot \cos x} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \color{blue}{\left(\left(1 + \tan x \cdot \tan \varepsilon\right) - 1\right)}\right) \cdot \cos x} \]

10. Final simplification 99.7%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\cos x \cdot \left(1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)\right)} \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \tan x\\ \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, t\_0, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + t\_0\right)\right)\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (tan x))))
   (/
    (*
     eps
     (+
      (cos x)
      (+
       (/ (pow (sin x) 2.0) (cos x))
       (*
        (pow eps 2.0)
        (fma
         0.3333333333333333
         t_0
         (fma
          0.3333333333333333
          (cos x)
          (* (pow eps 2.0) (* 0.13333333333333333 (+ (cos x) t_0)))))))))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * tan(x);
	return (eps * (cos(x) + ((pow(sin(x), 2.0) / cos(x)) + (pow(eps, 2.0) * fma(0.3333333333333333, t_0, fma(0.3333333333333333, cos(x), (pow(eps, 2.0) * (0.13333333333333333 * (cos(x) + t_0))))))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * tan(x))
	return Float64(Float64(eps * Float64(cos(x) + Float64(Float64((sin(x) ^ 2.0) / cos(x)) + Float64((eps ^ 2.0) * fma(0.3333333333333333, t_0, fma(0.3333333333333333, cos(x), Float64((eps ^ 2.0) * Float64(0.13333333333333333 * Float64(cos(x) + t_0))))))))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.3333333333333333 * t$95$0 + N[(0.3333333333333333 * N[Cos[x], $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.13333333333333333 * N[(N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 99.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\cos x + \left({\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. distribute-lft-in 99.7%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{\left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

9. Applied egg-rr 99.7%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{\left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right) + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
10. Step-by-step derivation

    1. +-commutative 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right) + {\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    2. distribute-lft-in 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{{\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right) + \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    3. fma-undefine 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\sin x}{1} \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    4. /-rgt-identity 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{\sin x} \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    5. fma-undefine 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \sin x \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \color{blue}{\left(\cos x \cdot 0.13333333333333333 + 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)}\right)\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    6. *-commutative 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \sin x \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\color{blue}{0.13333333333333333 \cdot \cos x} + 0.13333333333333333 \cdot \left(\frac{\sin x}{1} \cdot \tan x\right)\right)\right)\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

    7. distribute-lft-out 99.7%

       \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \sin x \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \color{blue}{\left(0.13333333333333333 \cdot \left(\cos x + \frac{\sin x}{1} \cdot \tan x\right)\right)}\right)\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

11. Simplified 99.7%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{{\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \sin x \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \sin x \cdot \tan x\right)\right)\right)\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

12. Final simplification 99.7%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \mathsf{fma}\left(0.3333333333333333, \sin x \cdot \tan x, \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \left(\cos x + \sin x \cdot \tan x\right)\right)\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
13. Add Preprocessing

Alternative 3: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \frac{\varepsilon \cdot \left(\cos x + \left(t\_0 + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot t\_0 + \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (cos x))))
   (/
    (*
     eps
     (+
      (cos x)
      (+
       t_0
       (*
        (pow eps 2.0)
        (+
         (* 0.3333333333333333 t_0)
         (fma
          0.3333333333333333
          (cos x)
          (* (pow eps 2.0) 0.13333333333333333)))))))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / cos(x);
	return (eps * (cos(x) + (t_0 + (pow(eps, 2.0) * ((0.3333333333333333 * t_0) + fma(0.3333333333333333, cos(x), (pow(eps, 2.0) * 0.13333333333333333))))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / cos(x))
	return Float64(Float64(eps * Float64(cos(x) + Float64(t_0 + Float64((eps ^ 2.0) * Float64(Float64(0.3333333333333333 * t_0) + fma(0.3333333333333333, cos(x), Float64((eps ^ 2.0) * 0.13333333333333333))))))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(0.3333333333333333 * t$95$0), $MachinePrecision] + N[(0.3333333333333333 * N[Cos[x], $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 99.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\cos x + \left({\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

8. Taylor expanded in x around 0 99.5%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, \color{blue}{0.13333333333333333 \cdot {\varepsilon}^{2}}\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
9. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, \color{blue}{{\varepsilon}^{2} \cdot 0.13333333333333333}\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

10. Simplified 99.5%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, \color{blue}{{\varepsilon}^{2} \cdot 0.13333333333333333}\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

11. Final simplification 99.5%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \left(0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x} + \mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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)
    code = (eps * (cos(x) + (t_0 + ((eps ** 2.0d0) * (0.3333333333333333d0 * (cos(x) + t_0)))))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 99.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\cos x + \left({\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

8. Taylor expanded in eps around 0 99.5%

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

   1. distribute-lft-out 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \left(0.3333333333333333 + {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (+
    (cos x)
    (+
     (/ (pow (sin x) 2.0) (cos x))
     (*
      (pow eps 2.0)
      (+ 0.3333333333333333 (* (pow eps 2.0) 0.13333333333333333))))))
  (* (cos x) (- 1.0 (* (tan x) (tan eps))))))

C

double code(double x, double eps) {
	return (eps * (cos(x) + ((pow(sin(x), 2.0) / cos(x)) + (pow(eps, 2.0) * (0.3333333333333333 + (pow(eps, 2.0) * 0.13333333333333333)))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (cos(x) + (((sin(x) ** 2.0d0) / cos(x)) + ((eps ** 2.0d0) * (0.3333333333333333d0 + ((eps ** 2.0d0) * 0.13333333333333333d0)))))) / (cos(x) * (1.0d0 - (tan(x) * tan(eps))))
end function

Java

public static double code(double x, double eps) {
	return (eps * (Math.cos(x) + ((Math.pow(Math.sin(x), 2.0) / Math.cos(x)) + (Math.pow(eps, 2.0) * (0.3333333333333333 + (Math.pow(eps, 2.0) * 0.13333333333333333)))))) / (Math.cos(x) * (1.0 - (Math.tan(x) * Math.tan(eps))));
}

Python

def code(x, eps):
	return (eps * (math.cos(x) + ((math.pow(math.sin(x), 2.0) / math.cos(x)) + (math.pow(eps, 2.0) * (0.3333333333333333 + (math.pow(eps, 2.0) * 0.13333333333333333)))))) / (math.cos(x) * (1.0 - (math.tan(x) * math.tan(eps))))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(cos(x) + Float64(Float64((sin(x) ^ 2.0) / cos(x)) + Float64((eps ^ 2.0) * Float64(0.3333333333333333 + Float64((eps ^ 2.0) * 0.13333333333333333)))))) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (cos(x) + (((sin(x) ^ 2.0) / cos(x)) + ((eps ^ 2.0) * (0.3333333333333333 + ((eps ^ 2.0) * 0.13333333333333333)))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[eps, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 99.7%

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\cos x + \left({\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\mathsf{fma}\left(0.3333333333333333, \cos x, {\varepsilon}^{2} \cdot \left(\cos x \cdot 0.13333333333333333 - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

8. Taylor expanded in x around 0 99.0%

   \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{{\varepsilon}^{2} \cdot \left(0.3333333333333333 + 0.13333333333333333 \cdot {\varepsilon}^{2}\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
9. Step-by-step derivation

   1. *-commutative 99.0%

      \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{\varepsilon}^{2} \cdot 0.13333333333333333}\right) - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

10. Simplified 99.0%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\color{blue}{{\varepsilon}^{2} \cdot \left(0.3333333333333333 + {\varepsilon}^{2} \cdot 0.13333333333333333\right)} - \left(-\frac{{\sin x}^{2}}{\cos x}\right)\right)\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

11. Final simplification 99.0%

    \[\leadsto \frac{\varepsilon \cdot \left(\cos x + \left(\frac{{\sin x}^{2}}{\cos x} + {\varepsilon}^{2} \cdot \left(0.3333333333333333 + {\varepsilon}^{2} \cdot 0.13333333333333333\right)\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
12. Add Preprocessing

Alternative 6: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-sum 63.0%

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

   2. tan-quot 63.0%

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

   3. frac-sub 63.0%

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

4. Applied egg-rr 63.0%

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

5. Taylor expanded in eps around 0 98.8%

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

   1. cancel-sign-sub-inv 98.8%

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

   2. metadata-eval 98.8%

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

   3. *-lft-identity 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 7: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\varepsilon + x\right) - \tan x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;e^{\log t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan (+ eps x)) (tan x))))
   (if (<= t_0 2e-14) eps (exp (log t_0)))))

C

double code(double x, double eps) {
	double t_0 = tan((eps + x)) - tan(x);
	double tmp;
	if (t_0 <= 2e-14) {
		tmp = eps;
	} else {
		tmp = exp(log(t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((eps + x)) - tan(x)
    if (t_0 <= 2d-14) then
        tmp = eps
    else
        tmp = exp(log(t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan((eps + x)) - Math.tan(x);
	double tmp;
	if (t_0 <= 2e-14) {
		tmp = eps;
	} else {
		tmp = Math.exp(Math.log(t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan((eps + x)) - math.tan(x)
	tmp = 0
	if t_0 <= 2e-14:
		tmp = eps
	else:
		tmp = math.exp(math.log(t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(Float64(eps + x)) - tan(x))
	tmp = 0.0
	if (t_0 <= 2e-14)
		tmp = eps;
	else
		tmp = exp(log(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan((eps + x)) - tan(x);
	tmp = 0.0;
	if (t_0 <= 2e-14)
		tmp = eps;
	else
		tmp = exp(log(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-14], eps, N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 2e-14

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

   if 2e-14 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 61.8%

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

      1. add-exp-log 61.9%

         \[\leadsto \color{blue}{e^{\log \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]

   4. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{e^{\log \left(\tan \left(x + \varepsilon\right) - \tan x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\tan \left(\varepsilon + x\right) - \tan x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (tan (+ eps x)) (tan x)) 2e-14)
   eps
   (- (/ 1.0 (/ (cos (+ eps x)) (sin (+ eps x)))) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if ((tan((eps + x)) - tan(x)) <= 2e-14) {
		tmp = eps;
	} else {
		tmp = (1.0 / (cos((eps + x)) / sin((eps + x)))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((tan((eps + x)) - tan(x)) <= 2d-14) then
        tmp = eps
    else
        tmp = (1.0d0 / (cos((eps + x)) / sin((eps + x)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.tan((eps + x)) - Math.tan(x)) <= 2e-14) {
		tmp = eps;
	} else {
		tmp = (1.0 / (Math.cos((eps + x)) / Math.sin((eps + x)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.tan((eps + x)) - math.tan(x)) <= 2e-14:
		tmp = eps
	else:
		tmp = (1.0 / (math.cos((eps + x)) / math.sin((eps + x)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(tan(Float64(eps + x)) - tan(x)) <= 2e-14)
		tmp = eps;
	else
		tmp = Float64(Float64(1.0 / Float64(cos(Float64(eps + x)) / sin(Float64(eps + x)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((tan((eps + x)) - tan(x)) <= 2e-14)
		tmp = eps;
	else
		tmp = (1.0 / (cos((eps + x)) / sin((eps + x)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 2e-14], eps, N[(N[(1.0 / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 2e-14

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

   if 2e-14 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 61.8%

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

      1. tan-quot 61.8%

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

      2. clear-num 61.8%

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

   4. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)}} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% 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 62.4%

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

3. Taylor expanded in eps around 0 97.6%

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

   1. sub-neg 97.6%

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

   2. mul-1-neg 97.6%

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

   3. remove-double-neg 97.6%

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

5. Simplified 97.6%

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

6. Final simplification 97.6%

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

Alternative 10: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2e-14) eps (- (tan (+ eps x)) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2e-14) {
		tmp = eps;
	} else {
		tmp = tan((eps + x)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2d-14) then
        tmp = eps
    else
        tmp = tan((eps + x)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2e-14) {
		tmp = eps;
	} else {
		tmp = Math.tan((eps + x)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2e-14:
		tmp = eps
	else:
		tmp = math.tan((eps + x)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2e-14)
		tmp = eps;
	else
		tmp = Float64(tan(Float64(eps + x)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2e-14)
		tmp = eps;
	else
		tmp = tan((eps + x)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2e-14], eps, N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2e-14

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

   if 2e-14 < eps

   1. Initial program 61.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% 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 62.4%

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

3. Taylor expanded in x around 0 95.8%

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

   1. tan-quot 95.8%

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

   2. *-un-lft-identity 95.8%

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

5. Applied egg-rr 95.8%

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

   1. *-lft-identity 95.8%

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

7. Simplified 95.8%

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

8. Final simplification 95.8%

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

Alternative 12: 97.6% 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 62.4%

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

3. Taylor expanded in x around 0 95.8%

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

4. Taylor expanded in eps around 0 95.8%

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

5. Final simplification 95.8%

   \[\leadsto \varepsilon \]
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 2024053 
(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)))