2tan (problem 3.3.2)

Percentage Accurate: 42.1% → 79.9%

Time: 18.8s

Alternatives: 12

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: 42.1% 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: 79.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

7. Taylor expanded in x around inf 67.5%

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

   1. associate--l+ 79.3%

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

   2. times-frac 79.3%

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

   3. associate-/r* 79.4%

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

   4. times-frac 79.4%

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

9. Simplified 79.4%

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

    1. tan-quot 77.7%

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

    2. sub-neg 77.7%

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

    3. tan-quot 79.4%

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

    4. tan-quot 79.4%

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

    5. tan-quot 79.4%

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

11. Applied egg-rr 79.4%

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

    1. sub-neg 79.4%

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

    2. *-commutative 79.4%

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

13. Simplified 79.4%

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

    1. tan-quot 79.5%

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

    2. associate-*r/ 79.5%

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

15. Applied egg-rr 79.5%

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

16. Final simplification 79.5%

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

Alternative 2: 79.9% accurate, 0.2× speedup

Math

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

FPCore

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

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(x) * tan(eps)))) - tan(x));
}

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(x) * tan(eps)))) - tan(x))
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(x) * Math.tan(eps)))) - Math.tan(x));
}

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(x) * math.tan(eps)))) - math.tan(x))

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(Float64(tan(x) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x)))
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(x) * tan(eps)))) - tan(x));
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[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

7. Taylor expanded in x around inf 67.5%

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

   1. associate--l+ 79.3%

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

   2. times-frac 79.3%

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

   3. associate-/r* 79.4%

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

   4. times-frac 79.4%

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

9. Simplified 79.4%

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

    1. tan-quot 77.7%

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

    2. sub-neg 77.7%

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

    3. tan-quot 79.4%

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

    4. tan-quot 79.4%

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

    5. tan-quot 79.4%

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

11. Applied egg-rr 79.4%

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

    1. sub-neg 79.4%

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

    2. *-commutative 79.4%

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

13. Simplified 79.4%

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

14. Taylor expanded in eps around inf 79.5%

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

    1. associate-/r* 79.4%

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

    2. times-frac 79.4%

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

16. Simplified 79.4%

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

17. Final simplification 79.4%

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

Alternative 3: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ \left(\frac{\tan x}{t\_0} - \tan x\right) + {\left(\sqrt[3]{\frac{\tan \varepsilon}{t\_0}}\right)}^{3} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps)))))
   (+ (- (/ (tan x) t_0) (tan x)) (pow (cbrt (/ (tan eps) t_0)) 3.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	return ((tan(x) / t_0) - tan(x)) + pow(cbrt((tan(eps) / t_0)), 3.0);
}

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 - (Math.tan(x) * Math.tan(eps));
	return ((Math.tan(x) / t_0) - Math.tan(x)) + Math.pow(Math.cbrt((Math.tan(eps) / t_0)), 3.0);
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	return Float64(Float64(Float64(tan(x) / t_0) - tan(x)) + (cbrt(Float64(tan(eps) / t_0)) ^ 3.0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[Tan[eps], $MachinePrecision] / t$95$0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

7. Taylor expanded in x around inf 67.5%

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

   1. associate--l+ 79.3%

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

   2. times-frac 79.3%

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

   3. associate-/r* 79.4%

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

   4. times-frac 79.4%

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

9. Simplified 79.4%

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

    1. tan-quot 77.7%

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

    2. sub-neg 77.7%

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

    3. tan-quot 79.4%

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

    4. tan-quot 79.4%

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

    5. tan-quot 79.4%

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

11. Applied egg-rr 79.4%

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

    1. sub-neg 79.4%

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

    2. *-commutative 79.4%

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

13. Simplified 79.4%

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

    1. add-cube-cbrt 78.3%

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

    2. pow3 78.3%

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

    3. associate-/r* 78.3%

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

    4. tan-quot 78.3%

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

    5. tan-quot 78.3%

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

    6. tan-quot 78.3%

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

15. Applied egg-rr 78.3%

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

16. Final simplification 78.3%

    \[\leadsto \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right) + {\left(\sqrt[3]{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}}\right)}^{3} \]
17. Add Preprocessing

Alternative 4: 65.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (- (- (tan x)) (tan eps)) (fma (tan x) (tan eps) -1.0)) (tan x)))

C

double code(double x, double eps) {
	return ((-tan(x) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x))
end

Wolfram

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

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

   1. frac-2neg 67.8%

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

   2. distribute-frac-neg 67.8%

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

   3. sub-neg 67.8%

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

   4. distribute-neg-in 67.8%

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

   5. metadata-eval 67.8%

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

   6. distribute-lft-neg-in 67.8%

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

   7. add-sqr-sqrt 33.7%

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

   8. sqrt-unprod 55.8%

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

   9. sqr-neg 55.8%

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

   10. sqrt-unprod 22.0%

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

   11. add-sqr-sqrt 44.5%

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

   12. distribute-lft-neg-in 44.5%

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

   13. add-sqr-sqrt 22.5%

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

8. Applied egg-rr 67.8%

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

   1. distribute-neg-frac 67.8%

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

   2. +-commutative 67.8%

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

   3. fma-def 67.8%

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

10. Simplified 67.8%

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

11. Final simplification 67.8%

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

Alternative 5: 65.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

   1. tan-quot 67.8%

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

   2. clear-num 67.8%

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

   3. un-div-inv 67.8%

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

   4. clear-num 67.8%

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

   5. tan-quot 67.8%

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

8. Applied egg-rr 67.8%

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

9. Final simplification 67.8%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x \]
10. Add Preprocessing

Alternative 6: 65.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := 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]

Derivation

1. Initial program 42.9%

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

   1. tan-sum 67.8%

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

   2. div-inv 67.7%

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

   3. *-un-lft-identity 67.7%

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

   4. prod-diff 67.7%

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

   5. *-commutative 67.7%

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

   6. *-un-lft-identity 67.7%

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

   7. *-commutative 67.7%

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

   8. *-un-lft-identity 67.7%

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

4. Applied egg-rr 67.7%

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

   1. +-commutative 67.7%

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

   2. fma-udef 67.7%

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

   3. associate-+r+ 67.7%

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

   4. unsub-neg 67.7%

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

6. Simplified 67.8%

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

7. Final simplification 67.8%

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

Alternative 7: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma eps (pow (tan x) 2.0) eps))

C

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

Julia

function code(x, eps)
	return fma(eps, (tan(x) ^ 2.0), eps)
end

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Taylor expanded in eps around 0 49.4%

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

   1. cancel-sign-sub-inv 49.4%

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

   2. metadata-eval 49.4%

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

   3. *-lft-identity 49.4%

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

5. Simplified 49.4%

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

   1. distribute-rgt-in 49.5%

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

   2. *-un-lft-identity 49.5%

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

   3. unpow2 49.5%

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

   4. unpow2 49.5%

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

   5. frac-times 49.4%

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

   6. tan-quot 49.5%

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

   7. tan-quot 49.5%

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

   8. pow2 49.5%

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

7. Applied egg-rr 49.5%

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

   1. +-commutative 49.5%

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

   2. *-commutative 49.5%

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

   3. fma-def 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

9. Simplified 49.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

10. Final simplification 49.5%

    \[\leadsto \mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right) \]
11. Add Preprocessing

Alternative 8: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Taylor expanded in eps around 0 49.4%

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

   1. cancel-sign-sub-inv 49.4%

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

   2. metadata-eval 49.4%

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

   3. *-lft-identity 49.4%

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

5. Simplified 49.4%

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

   1. *-un-lft-identity 49.4%

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

   2. unpow2 49.4%

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

   3. unpow2 49.4%

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

   4. frac-times 49.4%

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

   5. tan-quot 49.4%

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

   6. tan-quot 49.4%

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

   7. pow2 49.4%

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

7. Applied egg-rr 49.4%

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

   1. *-lft-identity 49.4%

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

9. Simplified 49.4%

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

10. Final simplification 49.4%

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

Alternative 9: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Taylor expanded in eps around 0 49.4%

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

   1. cancel-sign-sub-inv 49.4%

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

   2. metadata-eval 49.4%

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

   3. *-lft-identity 49.4%

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

5. Simplified 49.4%

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

   1. distribute-rgt-in 49.5%

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

   2. *-un-lft-identity 49.5%

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

   3. unpow2 49.5%

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

   4. unpow2 49.5%

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

   5. frac-times 49.4%

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

   6. tan-quot 49.5%

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

   7. tan-quot 49.5%

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

   8. pow2 49.5%

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

7. Applied egg-rr 49.5%

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

8. Final simplification 49.5%

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

Alternative 10: 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 42.9%

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

3. Taylor expanded in x around 0 56.4%

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

   1. tan-quot 56.5%

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

   2. expm1-log1p-u 51.3%

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

   3. expm1-udef 27.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]

5. Applied egg-rr 27.3%

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

   1. expm1-def 51.3%

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

   2. expm1-log1p 56.5%

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

7. Simplified 56.5%

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

8. Final simplification 56.5%

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

Alternative 11: 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 42.9%

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

3. Taylor expanded in x around 0 56.4%

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

   1. tan-quot 56.5%

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

   2. add-cbrt-cube 41.7%

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

   3. pow1/3 26.3%

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

   4. pow3 26.3%

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

5. Applied egg-rr 26.3%

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

   1. unpow1/3 41.7%

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

   2. rem-cbrt-cube 56.5%

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

   3. tan-quot 56.4%

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

   4. clear-num 56.3%

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

   5. clear-num 56.3%

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

   6. tan-quot 56.4%

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

7. Applied egg-rr 56.4%

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

8. Taylor expanded in eps around 0 29.7%

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

9. Final simplification 29.7%

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

Alternative 12: 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 42.9%

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

3. Taylor expanded in x around 0 56.4%

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

4. Taylor expanded in eps around 0 29.4%

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

5. Final simplification 29.4%

   \[\leadsto \varepsilon \]
6. 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 2024033 
(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)))