2tan (problem 3.3.2)

Percentage Accurate: 42.0% → 99.5%

Time: 18.3s

Alternatives: 9

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: 42.0% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(eps) / cos(eps)
    code = (t_0 / (1.0d0 - (t_0 * (sin(x) / cos(x))))) + (sin(x) / (cos(x) * (((1.0d0 / tan(eps)) / tan(x)) + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. tan-sum 68.4%

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

   2. div-inv 68.4%

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

   3. *-un-lft-identity 68.4%

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

   4. prod-diff 68.4%

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

   5. *-commutative 68.4%

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

   6. *-un-lft-identity 68.4%

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

   7. *-commutative 68.4%

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

   8. *-un-lft-identity 68.4%

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

4. Applied egg-rr 68.4%

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

6. Simplified 68.4%

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

7. Taylor expanded in x around inf 68.2%

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

   1. associate--l+ 81.7%

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

   2. associate-/r* 81.7%

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

   3. times-frac 81.7%

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

9. Simplified 81.8%

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

11. Applied egg-rr 79.3%

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

    1. associate-/r/ 79.3%

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

    2. rgt-mult-inverse 81.8%

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

    3. associate--r- 99.5%

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

    4. metadata-eval 99.5%

       \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\color{blue}{0} + \tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \tan x \]

    5. +-lft-identity 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

13. Simplified 99.5%

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

    1. clear-num 99.5%

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

    2. tan-quot 99.5%

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

    3. frac-times 99.5%

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

    4. *-un-lft-identity 99.5%

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

15. Applied egg-rr 99.5%

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

    1. *-commutative 99.5%

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

    2. div-sub 89.0%

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

    3. sub-neg 89.0%

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

    4. *-commutative 89.0%

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

    5. associate-/r* 89.0%

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

    6. *-inverses 99.6%

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

    7. metadata-eval 99.6%

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

17. Simplified 99.6%

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

18. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\tan x}{\frac{\frac{1}{\tan \varepsilon}}{\tan x} + -1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(eps) / cos(eps)
    code = (t_0 / (1.0d0 - (t_0 * (sin(x) / cos(x))))) + (tan(x) / (((1.0d0 / tan(eps)) / tan(x)) + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. tan-sum 68.4%

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

   2. div-inv 68.4%

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

   3. *-un-lft-identity 68.4%

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

   4. prod-diff 68.4%

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

   5. *-commutative 68.4%

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

   6. *-un-lft-identity 68.4%

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

   7. *-commutative 68.4%

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

   8. *-un-lft-identity 68.4%

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

4. Applied egg-rr 68.4%

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

6. Simplified 68.4%

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

7. Taylor expanded in x around inf 68.2%

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

   1. associate--l+ 81.7%

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

   2. associate-/r* 81.7%

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

   3. times-frac 81.7%

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

9. Simplified 81.8%

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

11. Applied egg-rr 79.3%

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

    1. associate-/r/ 79.3%

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

    2. rgt-mult-inverse 81.8%

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

    3. associate--r- 99.5%

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

    4. metadata-eval 99.5%

       \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\color{blue}{0} + \tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \tan x \]

    5. +-lft-identity 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

13. Simplified 99.5%

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

    1. expm1-log1p-u 92.9%

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

    2. expm1-udef 75.2%

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

    3. associate-*l/ 75.2%

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

    4. *-commutative 75.2%

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

    5. associate-*l* 75.2%

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

    6. pow2 75.2%

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

15. Applied egg-rr 75.2%

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

    1. expm1-def 92.9%

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

    2. expm1-log1p 99.5%

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

    3. *-commutative 99.5%

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

    4. unpow2 99.5%

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

    5. associate-*l* 99.5%

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

    6. associate-*l/ 99.5%

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

    7. associate-/r/ 99.6%

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

    8. div-sub 89.0%

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

    9. sub-neg 89.0%

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

    10. *-commutative 89.0%

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

    11. associate-/r* 89.0%

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

    12. *-inverses 99.5%

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

    13. metadata-eval 99.5%

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

17. Simplified 99.5%

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

18. Final simplification 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\tan x}{\frac{\frac{1}{\tan \varepsilon}}{\tan x} + -1} \]
19. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon \cdot \tan x\\ t_1 := 1 - t_0\\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{t_1} + \tan x \cdot \frac{t_0}{t_1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = tan(eps) * tan(x)
    t_1 = 1.0d0 - t_0
    code = ((sin(eps) / cos(eps)) / t_1) + (tan(x) * (t_0 / t_1))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.tan(eps) * math.tan(x)
	t_1 = 1.0 - t_0
	return ((math.sin(eps) / math.cos(eps)) / t_1) + (math.tan(x) * (t_0 / t_1))

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = tan(eps) * tan(x);
	t_1 = 1.0 - t_0;
	tmp = ((sin(eps) / cos(eps)) / t_1) + (tan(x) * (t_0 / t_1));
end

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. tan-sum 68.4%

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

   2. div-inv 68.4%

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

   3. *-un-lft-identity 68.4%

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

   4. prod-diff 68.4%

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

   5. *-commutative 68.4%

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

   6. *-un-lft-identity 68.4%

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

   7. *-commutative 68.4%

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

   8. *-un-lft-identity 68.4%

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

4. Applied egg-rr 68.4%

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

6. Simplified 68.4%

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

7. Taylor expanded in x around inf 68.2%

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

   1. associate--l+ 81.7%

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

   2. associate-/r* 81.7%

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

   3. times-frac 81.7%

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

9. Simplified 81.8%

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

11. Applied egg-rr 79.3%

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

    1. associate-/r/ 79.3%

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

    2. rgt-mult-inverse 81.8%

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

    3. associate--r- 99.5%

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

    4. metadata-eval 99.5%

       \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\color{blue}{0} + \tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \tan x \]

    5. +-lft-identity 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

13. Simplified 99.5%

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

    1. tan-quot 99.6%

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

    2. associate-*l/ 99.5%

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

    3. *-commutative 99.5%

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

    4. clear-num 99.5%

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

    5. clear-num 99.5%

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

    6. *-commutative 99.5%

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

    7. associate-*l/ 99.5%

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

    8. tan-quot 99.5%

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

15. Applied egg-rr 99.5%

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

    1. remove-double-div 99.5%

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

    2. *-commutative 99.5%

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

17. Simplified 99.5%

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

18. Final simplification 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan \varepsilon \cdot \tan x} + \tan x \cdot \frac{\tan \varepsilon \cdot \tan x}{1 - \tan \varepsilon \cdot \tan x} \]
19. Add Preprocessing

Alternative 4: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.6e-9) (not (<= eps 1.15e-16)))
   (- (/ (- (- (tan eps)) (tan x)) (fma (tan x) (tan eps) -1.0)) (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 <= -2.6e-9) || !(eps <= 1.15e-16)) {
		tmp = ((-tan(eps) - tan(x)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.6e-9) || !(eps <= 1.15e-16))
		tmp = Float64(Float64(Float64(Float64(-tan(eps)) - tan(x)) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.6e-9], N[Not[LessEqual[eps, 1.15e-16]], $MachinePrecision]], N[(N[(N[((-N[Tan[eps], $MachinePrecision]) - N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $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 < -2.6000000000000001e-9 or 1.15e-16 < eps

   1. Initial program 55.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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 99.2%

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

      3. *-un-lft-identity 99.2%

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

      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%

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

   4. Applied egg-rr 99.2%

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

   6. Simplified 99.2%

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

      1. frac-2neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. distribute-lft-neg-in 99.2%

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

      7. add-sqr-sqrt 55.1%

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

      8. sqrt-unprod 79.6%

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

      9. sqr-neg 79.6%

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

      10. sqrt-prod 24.5%

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

      11. add-sqr-sqrt 57.9%

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

      12. distribute-lft-neg-in 57.9%

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

      13. add-sqr-sqrt 33.3%

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

   8. Applied egg-rr 99.2%

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

      1. distribute-neg-frac 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.3%

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

   10. Simplified 99.3%

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

   if -2.6000000000000001e-9 < eps < 1.15e-16

   1. Initial program 30.7%

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

   3. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-9) (not (<= eps 1.15e-16)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan 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 <= -3.3e-9) || !(eps <= 1.15e-16)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(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 <= (-3.3d-9)) .or. (.not. (eps <= 1.15d-16))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(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 <= -3.3e-9) || !(eps <= 1.15e-16)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(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 <= -3.3e-9) or not (eps <= 1.15e-16):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(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 <= -3.3e-9) || !(eps <= 1.15e-16))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(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 <= -3.3e-9) || ~((eps <= 1.15e-16)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(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, -3.3e-9], N[Not[LessEqual[eps, 1.15e-16]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[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 < -3.30000000000000018e-9 or 1.15e-16 < eps

   1. Initial program 55.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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 99.2%

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

      3. *-un-lft-identity 99.2%

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

      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%

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

   4. Applied egg-rr 99.2%

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

   6. Simplified 99.2%

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

   if -3.30000000000000018e-9 < eps < 1.15e-16

   1. Initial program 30.7%

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

   3. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.32 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.32e-5) (not (<= eps 1.15e-16)))
   (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 <= -1.32e-5) || !(eps <= 1.15e-16)) {
		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 <= (-1.32d-5)) .or. (.not. (eps <= 1.15d-16))) 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 <= -1.32e-5) || !(eps <= 1.15e-16)) {
		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 <= -1.32e-5) or not (eps <= 1.15e-16):
		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 <= -1.32e-5) || !(eps <= 1.15e-16))
		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 <= -1.32e-5) || ~((eps <= 1.15e-16)))
		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, -1.32e-5], N[Not[LessEqual[eps, 1.15e-16]], $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 < -1.32000000000000007e-5 or 1.15e-16 < eps

   1. Initial program 55.7%

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

      1. log1p-expm1-u 54.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

   4. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

   5. Taylor expanded in x around 0 55.2%

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

      1. expm1-def 56.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]

   7. Simplified 56.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 23.4%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right)\right) \]

      2. sqrt-unprod 25.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right)\right) \]

      3. pow2 25.9%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{2}}}\right)\right) \]

      4. tan-quot 26.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{{\color{blue}{\tan \varepsilon}}^{2}}\right)\right) \]

   9. Applied egg-rr 26.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{{\tan \varepsilon}^{2}}}\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 26.0%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\tan \varepsilon \cdot \tan \varepsilon}}\right)\right) \]

       2. rem-sqrt-square 26.0%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left|\tan \varepsilon\right|}\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left|\tan \varepsilon\right|}\right)\right) \]

   12. Taylor expanded in eps around 0 26.0%

       \[\leadsto \color{blue}{\left|\tan \varepsilon\right|} \]
   13. Step-by-step derivation

       1. rem-square-sqrt 23.5%

          \[\leadsto \left|\color{blue}{\sqrt{\tan \varepsilon} \cdot \sqrt{\tan \varepsilon}}\right| \]

       2. fabs-sqr 23.5%

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

       3. rem-square-sqrt 58.1%

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

   14. Simplified 58.1%

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

   if -1.32000000000000007e-5 < eps < 1.15e-16

   1. Initial program 30.8%

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

   3. Taylor expanded in eps around 0 98.7%

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

      1. cancel-sign-sub-inv 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.32 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-16}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 44.2%

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

   1. log1p-expm1-u 43.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

4. Applied egg-rr 43.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

5. Taylor expanded in x around 0 33.0%

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

   1. expm1-def 58.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]

7. Simplified 58.4%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]
8. Step-by-step derivation

   1. add-sqr-sqrt 27.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right)\right) \]

   2. sqrt-unprod 22.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right)\right) \]

   3. pow2 22.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{2}}}\right)\right) \]

   4. tan-quot 22.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{{\color{blue}{\tan \varepsilon}}^{2}}\right)\right) \]

9. Applied egg-rr 22.7%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{{\tan \varepsilon}^{2}}}\right)\right) \]
10. Step-by-step derivation

    1. unpow2 22.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\tan \varepsilon \cdot \tan \varepsilon}}\right)\right) \]

    2. rem-sqrt-square 29.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left|\tan \varepsilon\right|}\right)\right) \]

11. Simplified 29.3%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left|\tan \varepsilon\right|}\right)\right) \]

12. Taylor expanded in eps around 0 29.3%

    \[\leadsto \color{blue}{\left|\tan \varepsilon\right|} \]
13. Step-by-step derivation

    1. rem-square-sqrt 27.1%

       \[\leadsto \left|\color{blue}{\sqrt{\tan \varepsilon} \cdot \sqrt{\tan \varepsilon}}\right| \]

    2. fabs-sqr 27.1%

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

    3. rem-square-sqrt 59.2%

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

14. Simplified 59.2%

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

15. Final simplification 59.2%

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

Alternative 8: 31.3% 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 44.2%

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

   1. log1p-expm1-u 43.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

4. Applied egg-rr 43.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

5. Taylor expanded in x around 0 33.0%

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

   1. expm1-def 58.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]

7. Simplified 58.4%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]
8. Step-by-step derivation

   1. log1p-expm1-u 59.1%

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

   2. clear-num 59.0%

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

   3. clear-num 58.9%

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

   4. tan-quot 59.1%

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

9. Applied egg-rr 59.1%

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

10. Taylor expanded in eps around 0 31.0%

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

11. Final simplification 31.0%

    \[\leadsto \frac{1}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}} \]
12. Add Preprocessing

Alternative 9: 30.9% 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 44.2%

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

   1. log1p-expm1-u 43.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

4. Applied egg-rr 43.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

5. Taylor expanded in x around 0 33.0%

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

   1. expm1-def 58.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]

7. Simplified 58.4%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right) \]
8. Step-by-step derivation

   1. log1p-expm1-u 59.1%

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

   2. clear-num 59.0%

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

   3. clear-num 58.9%

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

   4. tan-quot 59.1%

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

9. Applied egg-rr 59.1%

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

10. Taylor expanded in eps around 0 30.9%

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

11. Final simplification 30.9%

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

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 2024021 
(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)))