2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 100.0%

Time: 14.2s

Alternatives: 11

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon \cdot \cos x\\ t_1 := \sin \varepsilon \cdot \sin x\\ \frac{\frac{\sin \varepsilon}{\cos x}}{\frac{{t\_0}^{3} - {t\_1}^{3}}{\left(t\_1 \cdot t\_0 + {t\_1}^{2}\right) + {t\_0}^{2}}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos eps) (cos x))) (t_1 (* (sin eps) (sin x))))
   (/
    (/ (sin eps) (cos x))
    (/
     (- (pow t_0 3.0) (pow t_1 3.0))
     (+ (+ (* t_1 t_0) (pow t_1 2.0)) (pow t_0 2.0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) * cos(x);
	double t_1 = sin(eps) * sin(x);
	return (sin(eps) / cos(x)) / ((pow(t_0, 3.0) - pow(t_1, 3.0)) / (((t_1 * t_0) + pow(t_1, 2.0)) + pow(t_0, 2.0)));
}

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 = cos(eps) * cos(x)
    t_1 = sin(eps) * sin(x)
    code = (sin(eps) / cos(x)) / (((t_0 ** 3.0d0) - (t_1 ** 3.0d0)) / (((t_1 * t_0) + (t_1 ** 2.0d0)) + (t_0 ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) * Math.cos(x);
	double t_1 = Math.sin(eps) * Math.sin(x);
	return (Math.sin(eps) / Math.cos(x)) / ((Math.pow(t_0, 3.0) - Math.pow(t_1, 3.0)) / (((t_1 * t_0) + Math.pow(t_1, 2.0)) + Math.pow(t_0, 2.0)));
}

Python

def code(x, eps):
	t_0 = math.cos(eps) * math.cos(x)
	t_1 = math.sin(eps) * math.sin(x)
	return (math.sin(eps) / math.cos(x)) / ((math.pow(t_0, 3.0) - math.pow(t_1, 3.0)) / (((t_1 * t_0) + math.pow(t_1, 2.0)) + math.pow(t_0, 2.0)))

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) * cos(x))
	t_1 = Float64(sin(eps) * sin(x))
	return Float64(Float64(sin(eps) / cos(x)) / Float64(Float64((t_0 ^ 3.0) - (t_1 ^ 3.0)) / Float64(Float64(Float64(t_1 * t_0) + (t_1 ^ 2.0)) + (t_0 ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(eps) * cos(x);
	t_1 = sin(eps) * sin(x);
	tmp = (sin(eps) / cos(x)) / (((t_0 ^ 3.0) - (t_1 ^ 3.0)) / (((t_1 * t_0) + (t_1 ^ 2.0)) + (t_0 ^ 2.0)));
end

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\color{blue}{\cos \left(\varepsilon + x\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \color{blue}{\left(\varepsilon + x\right)}} \]

   3. cos-sum N/A

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

   4. flip3-- N/A

      \[\leadsto \frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\color{blue}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\sin \varepsilon \cdot \sin x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right) + \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) + \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\color{blue}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\sin \varepsilon \cdot \sin x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right) + \left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) + \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)}}} \]

8. Applied rewrites 100.0%

   \[\leadsto \frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\color{blue}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon \cdot \cos x\right)}^{2} + \left({\left(\sin x \cdot \sin \varepsilon\right)}^{2} + \left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)}}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\frac{\sin \varepsilon}{\cos x}}{\frac{{\left(\cos \varepsilon \cdot \cos x\right)}^{3} - {\left(\sin \varepsilon \cdot \sin x\right)}^{3}}{\left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right) + {\left(\sin \varepsilon \cdot \sin x\right)}^{2}\right) + {\left(\cos \varepsilon \cdot \cos x\right)}^{2}}} \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 60.4

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. +-inverses N/A

      \[\leadsto \frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   9. +-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. lower-+.f64 99.7

       \[\leadsto \frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. +-lft-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 100.0

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

   14. lift-+.f64 N/A

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

   15. +-lft-identity 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

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(Float64(sin(eps) / 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[(N[Sin[eps], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   2. +-lft-identity 99.7

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos x}}{\cos \left(x + \varepsilon\right)}} \]
9. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (/ (sin eps) (cos (+ x eps))) (cos x)))

C

double code(double x, double eps) {
	return (sin(eps) / cos((x + eps))) / cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) / cos((x + eps))) / cos(x)
end function

Java

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

Python

def code(x, eps):
	return (math.sin(eps) / math.cos((x + eps))) / math.cos(x)

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) / cos(Float64(x + eps))) / cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(eps) / cos((x + eps))) / cos(x);
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(x + \varepsilon\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 5: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}}{\cos \left(\varepsilon + x\right)} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]

   5. lift-/.f64 99.7

      \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   7. +-lft-identity 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 6: 99.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (/ eps (cos x)) (cos (+ x eps))))

C

double code(double x, double eps) {
	return (eps / cos(x)) / cos((x + eps));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return (eps / math.cos(x)) / math.cos((x + eps))

Julia

function code(x, eps)
	return Float64(Float64(eps / cos(x)) / cos(Float64(x + eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps / cos(x)) / cos((x + eps));
end

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

    \[\leadsto \frac{\frac{\varepsilon}{\cos x}}{\cos \left(x + \varepsilon\right)} \]
11. Add Preprocessing

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.6

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

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 8: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}}{\cos \left(\varepsilon + x\right)} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   5. lift-cos.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\cos x}} \]

   6. cos-mult N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\frac{\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \left(\left(\varepsilon + x\right) - x\right)}{2}}} \]

   7. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \left(\left(\varepsilon + x\right) - x\right)} \cdot 2} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(0 + \varepsilon\right)}{\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \left(\left(\varepsilon + x\right) - x\right)} \cdot 2} \]

8. Applied rewrites 99.7%

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

9. Taylor expanded in eps around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-*.f64 98.6

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

11. Applied rewrites 98.6%

    \[\leadsto \color{blue}{\frac{\varepsilon}{\cos \left(2 \cdot x\right) + 1}} \cdot 2 \]
12. Add Preprocessing

Alternative 9: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \varepsilon, x \cdot x, \varepsilon\right), x \cdot x, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma (* 0.6666666666666666 eps) (* x x) eps) (* x x) eps))

C

double code(double x, double eps) {
	return fma(fma((0.6666666666666666 * eps), (x * x), eps), (x * x), eps);
}

Julia

function code(x, eps)
	return fma(fma(Float64(0.6666666666666666 * eps), Float64(x * x), eps), Float64(x * x), eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.6666666666666666 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $MachinePrecision] * N[(x * x), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]

7. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \varepsilon, x \cdot x, \varepsilon\right), \color{blue}{x \cdot x}, \varepsilon\right) \]
13. Add Preprocessing

Alternative 10: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 60.4

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 60.4

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

4. Applied rewrites 60.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 60.4

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

   9. lift--.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate--l+ N/A

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

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\sin \left(\varepsilon + \color{blue}{0}\right)}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

   14. lower-+.f64 99.7

       \[\leadsto \frac{\frac{\sin \color{blue}{\left(0 + \varepsilon\right)}}{\cos x}}{\cos \left(\varepsilon + x\right)} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(0 + \varepsilon\right)}{\cos x}}{\cos \left(\varepsilon + x\right)}} \]

7. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.6

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

9. Applied rewrites 98.6%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.1%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\varepsilon}, \varepsilon\right) \]
13. Add Preprocessing

Alternative 11: 97.8% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2}, \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333, \varepsilon, \varepsilon\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot x, \varepsilon, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, x \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

12. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\varepsilon \cdot x, \varepsilon, \varepsilon\right) \]
13. Step-by-step derivation

14. Applied rewrites 97.8%

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

15. Final simplification 97.8%

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

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

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