2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.6%

Time: 18.3s

Alternatives: 11

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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 + 1\\ \mathsf{fma}\left(\varepsilon, t\_1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{t\_1}}\right) + {\varepsilon}^{3} \cdot \left({\left(\mathsf{hypot}\left(\tan x, t\_0\right)\right)}^{2} - \left(0.16666666666666666 + \mathsf{fma}\left(-0.5, t\_1, t\_0 \cdot 0.16666666666666666\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (+ t_0 1.0)))
   (+
    (fma eps t_1 (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) t_1)))
    (*
     (pow eps 3.0)
     (-
      (pow (hypot (tan x) t_0) 2.0)
      (+ 0.16666666666666666 (fma -0.5 t_1 (* t_0 0.16666666666666666))))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 + 1.0;
	return fma(eps, t_1, (pow(eps, 2.0) / ((cos(x) / sin(x)) / t_1))) + (pow(eps, 3.0) * (pow(hypot(tan(x), t_0), 2.0) - (0.16666666666666666 + fma(-0.5, t_1, (t_0 * 0.16666666666666666)))));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 + 1.0)
	return Float64(fma(eps, t_1, Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / t_1))) + Float64((eps ^ 3.0) * Float64((hypot(tan(x), t_0) ^ 2.0) - Float64(0.16666666666666666 + fma(-0.5, t_1, Float64(t_0 * 0.16666666666666666))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, N[(N[(eps * t$95$1 + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[Power[N[Sqrt[N[Tan[x], $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] - N[(0.16666666666666666 + N[(-0.5 * t$95$1 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
4. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

   2. mul-1-neg 100.0%

      \[\leadsto \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r- 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)} \]
8. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   3. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   4. sqrt-div 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   5. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   6. sqrt-prod 48.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   8. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   9. sqrt-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   11. tan-quot 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot {\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]
10. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]
12. Step-by-step derivation

    1. expm1-log1p-u 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right)\right)}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

    2. expm1-udef 99.6%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right)} - 1}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

13. Applied egg-rr 99.6%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{e^{\mathsf{log1p}\left(\frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot \left(1 + {\tan x}^{2}\right)}}\right)} - 1}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]
14. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot \left(1 + {\tan x}^{2}\right)}}\right)\right)}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

    2. expm1-log1p 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{\frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot \left(1 + {\tan x}^{2}\right)}}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

    3. associate-/r* 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \frac{{\varepsilon}^{2}}{\color{blue}{\frac{\frac{\cos x}{\sin x}}{1 + {\tan x}^{2}}}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

15. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \color{blue}{\frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + {\tan x}^{2}}}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

16. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, {\tan x}^{2} + 1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{{\tan x}^{2} + 1}}\right) + {\varepsilon}^{3} \cdot \left({\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2} - \left(0.16666666666666666 + \mathsf{fma}\left(-0.5, {\tan x}^{2} + 1, {\tan x}^{2} \cdot 0.16666666666666666\right)\right)\right) \]
17. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, {\tan x}^{2} + 1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1}}{\sin x}}\right) - \left(-1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + {\varepsilon}^{3} \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (fma
   eps
   (+ (pow (tan x) 2.0) 1.0)
   (/
    (pow eps 2.0)
    (/ (/ (cos x) (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0)) (sin x))))
  (+
   (* -1.3333333333333333 (* (pow eps 3.0) (pow x 2.0)))
   (* (pow eps 3.0) -0.3333333333333333))))

C

double code(double x, double eps) {
	return fma(eps, (pow(tan(x), 2.0) + 1.0), (pow(eps, 2.0) / ((cos(x) / ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)) / sin(x)))) - ((-1.3333333333333333 * (pow(eps, 3.0) * pow(x, 2.0))) + (pow(eps, 3.0) * -0.3333333333333333));
}

Julia

function code(x, eps)
	return Float64(fma(eps, Float64((tan(x) ^ 2.0) + 1.0), Float64((eps ^ 2.0) / Float64(Float64(cos(x) / Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)) / sin(x)))) - Float64(Float64(-1.3333333333333333 * Float64((eps ^ 3.0) * (x ^ 2.0))) + Float64((eps ^ 3.0) * -0.3333333333333333)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / 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] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.3333333333333333 * N[(N[Power[eps, 3.0], $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
4. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

   2. mul-1-neg 100.0%

      \[\leadsto \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r- 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)} \]
8. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   3. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   4. sqrt-div 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   5. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   6. sqrt-prod 48.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   8. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   9. sqrt-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   11. tan-quot 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot {\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]
10. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

12. Taylor expanded in x around 0 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + {\tan x}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - \color{blue}{\left(-1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + -0.3333333333333333 \cdot {\varepsilon}^{3}\right)} \]

13. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, {\tan x}^{2} + 1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1}}{\sin x}}\right) - \left(-1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + {\varepsilon}^{3} \cdot -0.3333333333333333\right) \]
14. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
4. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

   2. mul-1-neg 100.0%

      \[\leadsto \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r- 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)} \]
8. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   3. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   4. sqrt-div 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   5. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   6. sqrt-prod 48.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   8. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   9. sqrt-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   11. tan-quot 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot {\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]
10. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

12. Taylor expanded in x around 0 99.9%

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

13. Final simplification 99.9%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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)) - (((eps ** 2.0d0) * (sin(x) * ((-1.0d0) - (tan(x) ** 2.0d0)))) / cos(x))
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)) - ((Math.pow(eps, 2.0) * (Math.sin(x) * (-1.0 - Math.pow(Math.tan(x), 2.0)))) / Math.cos(x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. expm1-log1p-u 99.5%

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

   2. expm1-udef 99.5%

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

5. Applied egg-rr 99.5%

   \[\leadsto \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x \cdot \left(1 + {\tan x}^{2}\right)\right)} - 1\right)}}{\cos x} \]
6. Step-by-step derivation

   1. expm1-def 99.5%

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

   2. expm1-log1p 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.9%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   3. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   4. sqrt-div 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   5. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   6. sqrt-prod 48.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   8. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   9. sqrt-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   11. tan-quot 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

5. Applied egg-rr 99.9%

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

   1. *-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 6: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \cdot \sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   3. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot \color{blue}{{\left(\sqrt{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right)}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   4. sqrt-div 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\left(\frac{\sqrt{{\sin x}^{2}}}{\sqrt{{\cos x}^{2}}}\right)}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   5. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sqrt{\color{blue}{\sin x \cdot \sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   6. sqrt-prod 48.8%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sqrt{\sin x} \cdot \sqrt{\sin x}}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\color{blue}{\sin x}}{\sqrt{{\cos x}^{2}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   8. unpow2 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\sqrt{\color{blue}{\cos x \cdot \cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   9. sqrt-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\sqrt{\cos x} \cdot \sqrt{\cos x}}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\left(\frac{\sin x}{\color{blue}{\cos x}}\right)}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

   11. tan-quot 100.0%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + 1 \cdot {\color{blue}{\tan x}}^{2}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

6. Applied egg-rr 99.6%

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

   1. *-lft-identity 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{{\tan x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) - {\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right) \]

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 7: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. *-lft-identity 99.5%

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

5. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. pow2 99.5%

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

   5. sqrt-div 99.5%

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

   6. unpow2 99.5%

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

   7. sqrt-prod 48.5%

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

   8. add-sqr-sqrt 99.5%

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

   9. unpow2 99.5%

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

   10. sqrt-prod 99.3%

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

   11. add-sqr-sqrt 99.5%

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

   12. tan-quot 99.5%

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

7. Applied egg-rr 99.5%

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

   1. distribute-rgt1-in 99.5%

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

   2. +-commutative 99.5%

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

   3. *-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 8: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. *-lft-identity 99.5%

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

5. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. pow2 99.5%

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

   5. sqrt-div 99.5%

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

   6. unpow2 99.5%

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

   7. sqrt-prod 48.5%

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

   8. add-sqr-sqrt 99.5%

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

   9. unpow2 99.5%

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

   10. sqrt-prod 99.3%

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

   11. add-sqr-sqrt 99.5%

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

   12. tan-quot 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 9: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. *-lft-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 10: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. *-lft-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

8. Simplified 98.6%

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

   1. distribute-rgt1-in 98.6%

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

   2. unpow2 98.6%

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

   3. fma-def 98.6%

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

10. Applied egg-rr 98.6%

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

11. Final simplification 98.6%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \]
12. Add Preprocessing

Alternative 11: 98.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. *-lft-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 98.1%

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

7. Final simplification 98.1%

   \[\leadsto \varepsilon \]
8. 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 2024026 
(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)))

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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