2tan (problem 3.3.2)

Percentage Accurate: 42.5% → 99.5%

Time: 18.5s

Alternatives: 14

Speedup: 2.0×

Specification

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: 42.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

   1. tan-sum 66.8%

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

   2. div-inv 66.8%

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

   3. *-un-lft-identity 66.8%

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

   4. *-commutative 66.8%

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

   5. prod-diff 66.8%

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

   6. *-un-lft-identity 66.8%

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

   7. metadata-eval 66.8%

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

   8. *-un-lft-identity 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x around inf 66.7%

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

6. Simplified 79.9%

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

   1. tan-quot 78.2%

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

   2. clear-num 76.9%

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

   3. frac-sub 76.2%

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

8. Applied egg-rr 77.6%

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

   1. rgt-mult-inverse 79.9%

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

   2. associate--r- 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-lft-identity 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

   1. tan-sum 66.8%

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

   2. div-inv 66.8%

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

   3. *-un-lft-identity 66.8%

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

   4. *-commutative 66.8%

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

   5. prod-diff 66.8%

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

   6. *-un-lft-identity 66.8%

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

   7. metadata-eval 66.8%

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

   8. *-un-lft-identity 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x around inf 66.7%

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

6. Simplified 79.9%

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

   1. tan-quot 78.2%

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

   2. clear-num 76.9%

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

   3. frac-sub 76.2%

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

8. Applied egg-rr 77.6%

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

   1. rgt-mult-inverse 79.9%

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

   2. associate--r- 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-lft-identity 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. clear-num 99.5%

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

    2. frac-times 99.5%

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

    3. *-un-lft-identity 99.5%

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

    4. clear-num 99.5%

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

    5. tan-quot 99.5%

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

12. Applied egg-rr 99.5%

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

    1. associate-/r* 99.5%

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

    2. associate-/l* 99.5%

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

    3. /-rgt-identity 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

   1. tan-sum 66.8%

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

   2. div-inv 66.8%

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

   3. *-un-lft-identity 66.8%

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

   4. *-commutative 66.8%

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

   5. prod-diff 66.8%

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

   6. *-un-lft-identity 66.8%

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

   7. metadata-eval 66.8%

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

   8. *-un-lft-identity 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x around inf 66.7%

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

6. Simplified 79.9%

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

   1. tan-quot 78.2%

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

   2. clear-num 76.9%

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

   3. frac-sub 76.2%

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

8. Applied egg-rr 77.6%

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

   1. rgt-mult-inverse 79.9%

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

   2. associate--r- 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-lft-identity 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. tan-quot 99.5%

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

    2. tan-quot 99.5%

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

    3. *-commutative 99.5%

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

    4. rem-exp-log 47.4%

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

    5. expm1-log1p-u 47.4%

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

    6. expm1-udef 47.4%

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

    7. log1p-udef 47.4%

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

    8. add-exp-log 47.4%

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

    9. rem-exp-log 99.5%

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

    10. *-commutative 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := 1 - t_0\\ \mathsf{fma}\left(-1, \tan x, \tan x\right) + \left(\frac{t_0}{\frac{t_1}{\tan x}} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot t_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (- 1.0 t_0)))
   (+
    (fma -1.0 (tan x) (tan x))
    (+ (/ t_0 (/ t_1 (tan x))) (/ (sin eps) (* (cos eps) t_1))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = 1.0 - t_0;
	return fma(-1.0, tan(x), tan(x)) + ((t_0 / (t_1 / tan(x))) + (sin(eps) / (cos(eps) * t_1)));
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(1.0 - t_0)
	return Float64(fma(-1.0, tan(x), tan(x)) + Float64(Float64(t_0 / Float64(t_1 / tan(x))) + Float64(sin(eps) / Float64(cos(eps) * t_1))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, N[(N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / N[(t$95$1 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 41.2%

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

   1. tan-sum 66.8%

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

   2. div-inv 66.8%

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

   3. *-un-lft-identity 66.8%

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

   4. *-commutative 66.8%

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

   5. prod-diff 66.8%

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

   6. *-un-lft-identity 66.8%

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

   7. metadata-eval 66.8%

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

   8. *-un-lft-identity 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x around inf 66.7%

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

6. Simplified 79.9%

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

   1. tan-quot 78.2%

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

   2. clear-num 76.9%

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

   3. frac-sub 76.2%

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

8. Applied egg-rr 77.6%

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

   1. rgt-mult-inverse 79.9%

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

   2. associate--r- 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-lft-identity 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. clear-num 99.5%

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

    2. clear-num 99.5%

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

    3. frac-times 99.5%

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

    4. metadata-eval 99.5%

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

    5. clear-num 99.5%

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

    6. tan-quot 99.5%

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

    7. clear-num 99.5%

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

    8. tan-quot 99.5%

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

12. Applied egg-rr 99.5%

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

    1. associate-/r* 99.5%

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

    2. remove-double-div 99.5%

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

    3. associate-/r/ 99.5%

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

    4. /-rgt-identity 99.5%

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

    5. *-commutative 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -6.6e-8)
     (- (/ t_0 (- 1.0 (/ (* (sin eps) (tan x)) (cos eps)))) (tan x))
     (if (<= eps 2.8e-7)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (pow eps 2.0)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (- (/ t_0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -6.6e-8) {
		tmp = (t_0 / (1.0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x);
	} else if (eps <= 2.8e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	}
	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(x) + tan(eps)
    if (eps <= (-6.6d-8)) then
        tmp = (t_0 / (1.0d0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x)
    else if (eps <= 2.8d-7) then
        tmp = (eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = (t_0 / (1.0d0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -6.6e-8) {
		tmp = (t_0 / (1.0 - ((Math.sin(eps) * Math.tan(x)) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 2.8e-7) {
		tmp = (eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + (Math.pow(eps, 2.0) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - ((Math.sin(x) * Math.tan(eps)) / Math.cos(x)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -6.6e-8:
		tmp = (t_0 / (1.0 - ((math.sin(eps) * math.tan(x)) / math.cos(eps)))) - math.tan(x)
	elif eps <= 2.8e-7:
		tmp = (eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + (math.pow(eps, 2.0) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = (t_0 / (1.0 - ((math.sin(x) * math.tan(eps)) / math.cos(x)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -6.6e-8)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(eps) * tan(x)) / cos(eps)))) - tan(x));
	elseif (eps <= 2.8e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -6.6e-8)
		tmp = (t_0 / (1.0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x);
	elseif (eps <= 2.8e-7)
		tmp = (eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps ^ 2.0) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.6e-8], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-7], N[(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] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.59999999999999954e-8

   1. Initial program 52.6%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.4%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

   if -6.59999999999999954e-8 < eps < 2.80000000000000019e-7

   1. Initial program 29.7%

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

      1. tan-sum 30.1%

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

      2. div-inv 30.1%

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

      3. *-un-lft-identity 30.1%

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

      4. prod-diff 30.1%

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

      5. *-commutative 30.1%

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

      6. *-un-lft-identity 30.1%

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

      7. *-commutative 30.1%

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

      8. *-un-lft-identity 30.1%

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

   3. Applied egg-rr 30.1%

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

      1. +-commutative 30.1%

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

      2. fma-udef 30.1%

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

      3. associate-+r+ 30.1%

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

      4. unsub-neg 30.1%

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

   5. Simplified 30.1%

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

      1. add-exp-log 20.9%

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

   7. Applied egg-rr 20.9%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-lft-identity 99.6%

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

   10. Simplified 99.6%

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

   if 2.80000000000000019e-7 < eps

   1. Initial program 49.8%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. tan-quot 99.1%

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

      3. associate-*r/ 99.1%

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

   7. Applied egg-rr 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4e-9) (not (<= eps 4.8e-9)))
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (/ (* (sin x) (tan eps)) (cos x))))
    (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e-9) || !(eps <= 4.8e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4d-9)) .or. (.not. (eps <= 4.8d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e-9) || !(eps <= 4.8e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - ((Math.sin(x) * Math.tan(eps)) / Math.cos(x)))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4e-9) or not (eps <= 4.8e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - ((math.sin(x) * math.tan(eps)) / math.cos(x)))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4e-9) || !(eps <= 4.8e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4e-9) || ~((eps <= 4.8e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4e-9], N[Not[LessEqual[eps, 4.8e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if eps < -4.00000000000000025e-9 or 4.8e-9 < eps

   1. Initial program 51.4%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. tan-quot 99.3%

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

      3. associate-*r/ 99.3%

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

   7. Applied egg-rr 99.3%

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

   if -4.00000000000000025e-9 < eps < 4.8e-9

   1. Initial program 29.7%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4e-9)
     (- (/ t_0 (- 1.0 (/ (* (sin eps) (tan x)) (cos eps)))) (tan x))
     (if (<= eps 4.8e-9)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ t_0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4e-9) {
		tmp = (t_0 / (1.0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x);
	} else if (eps <= 4.8e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	}
	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(x) + tan(eps)
    if (eps <= (-4d-9)) then
        tmp = (t_0 / (1.0d0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x)
    else if (eps <= 4.8d-9) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 / (1.0d0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -4e-9) {
		tmp = (t_0 / (1.0 - ((Math.sin(eps) * Math.tan(x)) / Math.cos(eps)))) - Math.tan(x);
	} else if (eps <= 4.8e-9) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 / (1.0 - ((Math.sin(x) * Math.tan(eps)) / Math.cos(x)))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -4e-9:
		tmp = (t_0 / (1.0 - ((math.sin(eps) * math.tan(x)) / math.cos(eps)))) - math.tan(x)
	elif eps <= 4.8e-9:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 / (1.0 - ((math.sin(x) * math.tan(eps)) / math.cos(x)))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(eps) * tan(x)) / cos(eps)))) - tan(x));
	elseif (eps <= 4.8e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -4e-9)
		tmp = (t_0 / (1.0 - ((sin(eps) * tan(x)) / cos(eps)))) - tan(x);
	elseif (eps <= 4.8e-9)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.8e-9], 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], N[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000025e-9

   1. Initial program 52.6%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.4%

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

      1. tan-quot 99.4%

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

      2. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

   if -4.00000000000000025e-9 < eps < 4.8e-9

   1. Initial program 29.7%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   4. Simplified 99.4%

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

   if 4.8e-9 < eps

   1. Initial program 49.8%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

      5. *-commutative 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-un-lft-identity 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. tan-quot 99.1%

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

      3. associate-*r/ 99.1%

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

   7. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array} \]

Alternative 8: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{1}{1 - \tan x \cdot \tan \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -5.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1 - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (/ 1.0 (- 1.0 (* (tan x) (tan eps))))))
   (if (<= eps -5.6e-9)
     (fma t_0 t_1 (- (tan x)))
     (if (<= eps 5e-9)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 / (1.0 - (tan(x) * tan(eps)));
	double tmp;
	if (eps <= -5.6e-9) {
		tmp = fma(t_0, t_1, -tan(x));
	} else if (eps <= 5e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * t_1) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))
	tmp = 0.0
	if (eps <= -5.6e-9)
		tmp = fma(t_0, t_1, Float64(-tan(x)));
	elseif (eps <= 5e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * t_1) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.6e-9], N[(t$95$0 * t$95$1 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 5e-9], 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], N[(N[(t$95$0 * t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.59999999999999969e-9

   1. Initial program 52.6%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. prod-diff 99.4%

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

      5. *-commutative 99.4%

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

      6. *-un-lft-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. *-un-lft-identity 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.4%

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

      3. associate-+r+ 99.4%

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

      4. unsub-neg 99.4%

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

   5. Simplified 99.4%

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

      1. add-exp-log 36.5%

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

   7. Applied egg-rr 36.5%

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

      1. add-exp-log 99.4%

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

      2. *-commutative 99.4%

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

      3. tan-quot 99.4%

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

      4. tan-quot 99.4%

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

      5. div-inv 99.5%

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

      6. fma-neg 99.5%

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

   9. Applied egg-rr 99.4%

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

       1. +-commutative 99.4%

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

       2. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if -5.59999999999999969e-9 < eps < 5.0000000000000001e-9

   1. Initial program 29.7%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   4. Simplified 99.4%

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

   if 5.0000000000000001e-9 < eps

   1. Initial program 49.8%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.25 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.25e-9) (not (<= eps 5e-9)))
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.25e-9) || !(eps <= 5e-9)) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.25d-9)) .or. (.not. (eps <= 5d-9))) then
        tmp = ((tan(x) + tan(eps)) * (1.0d0 / (1.0d0 - (tan(x) * tan(eps))))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.25e-9) || !(eps <= 5e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) * (1.0 / (1.0 - (Math.tan(x) * Math.tan(eps))))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.25e-9) or not (eps <= 5e-9):
		tmp = ((math.tan(x) + math.tan(eps)) * (1.0 / (1.0 - (math.tan(x) * math.tan(eps))))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.25e-9) || !(eps <= 5e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) * Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.25e-9) || ~((eps <= 5e-9)))
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.25e-9], N[Not[LessEqual[eps, 5e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if eps < -3.2500000000000002e-9 or 5.0000000000000001e-9 < eps

   1. Initial program 51.4%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   if -3.2500000000000002e-9 < eps < 5.0000000000000001e-9

   1. Initial program 29.7%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.25 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 10: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.4e-9) (not (<= eps 4.8e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-9) || !(eps <= 4.8e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4.4d-9)) .or. (.not. (eps <= 4.8d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.4e-9) || !(eps <= 4.8e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.4e-9) or not (eps <= 4.8e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.4e-9) || !(eps <= 4.8e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.4e-9) || ~((eps <= 4.8e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.4e-9], N[Not[LessEqual[eps, 4.8e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if eps < -4.3999999999999997e-9 or 4.8e-9 < eps

   1. Initial program 51.4%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

      4. unsub-neg 99.3%

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

   5. Simplified 99.2%

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

   if -4.3999999999999997e-9 < eps < 4.8e-9

   1. Initial program 29.7%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 11: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.00012\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -6.6e-8) (not (<= eps 0.00012)))
   (tan eps)
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.6e-8) || !(eps <= 0.00012)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-6.6d-8)) .or. (.not. (eps <= 0.00012d0))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -6.6e-8) || !(eps <= 0.00012)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -6.6e-8) or not (eps <= 0.00012):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -6.6e-8) || !(eps <= 0.00012))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -6.6e-8) || ~((eps <= 0.00012)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -6.6e-8], N[Not[LessEqual[eps, 0.00012]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], 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. Split input into 2 regimes

2. if eps < -6.59999999999999954e-8 or 1.20000000000000003e-4 < eps

   1. Initial program 51.7%

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

   2. Taylor expanded in x around 0 55.1%

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

      1. tan-quot 55.4%

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

   4. Applied egg-rr 55.4%

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

   if -6.59999999999999954e-8 < eps < 1.20000000000000003e-4

   1. Initial program 29.4%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.6 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.00012\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 12: 57.9% 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 41.2%

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

2. Taylor expanded in x around 0 56.3%

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

   1. tan-quot 56.5%

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

4. Applied egg-rr 56.5%

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

5. Final simplification 56.5%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 31.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ 1.0 (+ (* eps -0.3333333333333333) (/ 1.0 eps))))

C

double code(double x, double eps) {
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
}

Python

def code(x, eps):
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps))

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(eps * -0.3333333333333333) + Float64(1.0 / eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(eps * -0.3333333333333333), $MachinePrecision] + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.2%

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

2. Taylor expanded in x around 0 56.3%

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

   1. add-cbrt-cube 40.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}} \]

   2. pow1/3 21.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{0.3333333333333333}} \]

   3. tan-quot 21.6%

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

   4. tan-quot 21.6%

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

   5. tan-quot 21.6%

      \[\leadsto {\left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \color{blue}{\tan \varepsilon}\right)}^{0.3333333333333333} \]

   6. pow3 21.6%

      \[\leadsto {\color{blue}{\left({\tan \varepsilon}^{3}\right)}}^{0.3333333333333333} \]

4. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{{\left({\tan \varepsilon}^{3}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 40.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\tan \varepsilon}^{3}}} \]

   2. rem-cbrt-cube 56.5%

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

   3. tan-quot 56.3%

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

   4. clear-num 56.2%

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

   5. clear-num 56.2%

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

   6. tan-quot 56.3%

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

6. Applied egg-rr 56.3%

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

7. Taylor expanded in eps around 0 29.7%

   \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot \varepsilon + \frac{1}{\varepsilon}}} \]

8. Final simplification 29.7%

   \[\leadsto \frac{1}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}} \]

Alternative 14: 30.6% 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 41.2%

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

2. Taylor expanded in x around 0 56.3%

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

3. Taylor expanded in eps around 0 29.3%

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

4. Final simplification 29.3%

   \[\leadsto \varepsilon \]

Developer target: 76.3% 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 2023339 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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