2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.7%

Time: 15.0s

Alternatives: 12

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \mathsf{fma}\left(t\_0, 0.3333333333333333, 0.3333333333333333\right)\right), \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (fma
    (fma
     (fma
      eps
      (fma
       (tan x)
       (tan x)
       (+ (pow (tan x) 4.0) (fma t_0 0.3333333333333333 0.3333333333333333)))
      (* (fma (tan x) (tan x) 1.0) (tan x)))
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return fma(fma(fma(eps, fma(tan(x), tan(x), (pow(tan(x), 4.0) + fma(t_0, 0.3333333333333333, 0.3333333333333333))), (fma(tan(x), tan(x), 1.0) * tan(x))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return fma(fma(fma(eps, fma(tan(x), tan(x), Float64((tan(x) ^ 4.0) + fma(t_0, 0.3333333333333333, 0.3333333333333333))), Float64(fma(tan(x), tan(x), 1.0) * tan(x))), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(eps * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision] + N[(t$95$0 * 0.3333333333333333 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left(\sin x, \sin x, {\left(\sin x \cdot \tan x\right)}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]
10. Step-by-step derivation

11. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}\right), \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \color{blue}{\varepsilon}, \varepsilon\right) \]

12. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right)\right), \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 0.3333333333333333 + {\tan x}^{4}\right), \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    (fma (tan x) (tan x) (+ 0.3333333333333333 (pow (tan x) 4.0)))
    eps
    (/ (* (sin x) (fma (tan x) (tan x) 1.0)) (cos x)))
   eps
   (pow (tan x) 2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma(tan(x), tan(x), (0.3333333333333333 + pow(tan(x), 4.0))), eps, ((sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(tan(x), tan(x), Float64(0.3333333333333333 + (tan(x) ^ 4.0))), eps, Float64(Float64(sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(0.3333333333333333 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left(\sin x, \sin x, {\left(\sin x \cdot \tan x\right)}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \frac{1}{3}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + 0.3333333333333333\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

13. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 0.3333333333333333 + {\tan x}^{4}\right), \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
14. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left(\sin x, \sin x, {\left(\sin x \cdot \tan x\right)}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3} + \frac{4}{3} \cdot {x}^{2}, \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left(\sin x, \sin x, {\left(\sin x \cdot \tan x\right)}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.2%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right), \varepsilon, \varepsilon\right) \]

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Add Preprocessing

Alternative 6: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.3333333333333333, 0.20833333333333334, 0.16666666666666666\right), x \cdot x, 0.3333333333333333\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right)\right) \cdot \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (fma
     (fma
      (* (* x x) 0.3333333333333333)
      0.20833333333333334
      0.16666666666666666)
     (* x x)
     0.3333333333333333)
    (* eps eps)
    (fma (fma (* x x) 0.20833333333333334 0.5) (* x x) 1.0))
   eps)
  (* (- 1.0 (* (tan eps) (tan x))) (cos x))))

C

double code(double x, double eps) {
	return (fma(fma(fma(((x * x) * 0.3333333333333333), 0.20833333333333334, 0.16666666666666666), (x * x), 0.3333333333333333), (eps * eps), fma(fma((x * x), 0.20833333333333334, 0.5), (x * x), 1.0)) * eps) / ((1.0 - (tan(eps) * tan(x))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(fma(Float64(Float64(x * x) * 0.3333333333333333), 0.20833333333333334, 0.16666666666666666), Float64(x * x), 0.3333333333333333), Float64(eps * eps), fma(fma(Float64(x * x), 0.20833333333333334, 0.5), Float64(x * x), 1.0)) * eps) / Float64(Float64(1.0 - Float64(tan(eps) * tan(x))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * 0.20833333333333334 + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] / N[(N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 63.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 63.2

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

4. Applied rewrites 63.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. remove-double-div 63.2

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

   5. lift--.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. frac-sub N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 63.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \frac{\sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot {x}^{2}} + \frac{\sin \varepsilon}{\cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}, {x}^{2}, \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

9. Applied rewrites 98.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot 0.20833333333333334, x \cdot x, \frac{0.5 \cdot \sin \varepsilon}{\cos \varepsilon}\right), x \cdot x, \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{6} + \frac{5}{24} \cdot \left(\frac{-1}{6} \cdot {x}^{2} - \frac{-1}{2} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{5}{24} \cdot {x}^{2}\right)\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
11. Step-by-step derivation

12. Applied rewrites 97.9%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.3333333333333333, 0.20833333333333334, 0.16666666666666666\right), x \cdot x, 0.3333333333333333\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right)\right) \cdot \color{blue}{\varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

13. Final simplification 97.9%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.3333333333333333, 0.20833333333333334, 0.16666666666666666\right), x \cdot x, 0.3333333333333333\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right)\right) \cdot \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x} \]
14. Add Preprocessing

Alternative 7: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right) \cdot \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma (fma (* x x) 0.20833333333333334 0.5) (* x x) 1.0) eps)
  (* (- 1.0 (* (tan eps) (tan x))) (cos x))))

C

double code(double x, double eps) {
	return (fma(fma((x * x), 0.20833333333333334, 0.5), (x * x), 1.0) * eps) / ((1.0 - (tan(eps) * tan(x))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(x * x), 0.20833333333333334, 0.5), Float64(x * x), 1.0) * eps) / Float64(Float64(1.0 - Float64(tan(eps) * tan(x))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.20833333333333334 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 63.2

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 63.2

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

4. Applied rewrites 63.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. remove-double-div 63.2

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

   5. lift--.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. tan-sum N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. frac-sub N/A

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

   14. lower-/.f64 N/A

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

6. Applied rewrites 63.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) + \frac{\sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot {x}^{2}} + \frac{\sin \varepsilon}{\cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} - \left(\frac{-1}{3} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \frac{1}{6} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)\right)\right) - -1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}, {x}^{2}, \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

9. Applied rewrites 98.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot 0.20833333333333334, x \cdot x, \frac{0.5 \cdot \sin \varepsilon}{\cos \varepsilon}\right), x \cdot x, \frac{\sin \varepsilon}{\cos \varepsilon}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{5}{24} \cdot {x}^{2}\right)\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
11. Step-by-step derivation

12. Applied rewrites 97.7%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right) \cdot \color{blue}{\varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]

13. Final simplification 97.7%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.20833333333333334, 0.5\right), x \cdot x, 1\right) \cdot \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x} \]
14. Add Preprocessing

Alternative 8: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right) \cdot x\right), x, \varepsilon \cdot \varepsilon\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333, \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    (fma 1.3333333333333333 (* eps eps) 1.0)
    eps
    (* (* (* eps eps) 1.3333333333333333) x))
   x
   (* eps eps))
  x
  (fma (pow eps 3.0) 0.3333333333333333 eps)))

C

double code(double x, double eps) {
	return fma(fma(fma(fma(1.3333333333333333, (eps * eps), 1.0), eps, (((eps * eps) * 1.3333333333333333) * x)), x, (eps * eps)), x, fma(pow(eps, 3.0), 0.3333333333333333, eps));
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(1.3333333333333333, Float64(eps * eps), 1.0), eps, Float64(Float64(Float64(eps * eps) * 1.3333333333333333) * x)), x, Float64(eps * eps)), x, fma((eps ^ 3.0), 0.3333333333333333, eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps + N[(N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[eps, 3.0], $MachinePrecision] * 0.3333333333333333 + eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 97.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \varepsilon, \left(1.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), x, \varepsilon \cdot \varepsilon\right), \color{blue}{x}, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333, \varepsilon\right)\right) \]

12. Final simplification 97.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 1.3333333333333333\right) \cdot x\right), x, \varepsilon \cdot \varepsilon\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333, \varepsilon\right)\right) \]
13. Add Preprocessing

Alternative 9: 98.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \left(x + \varepsilon\right) \cdot \varepsilon, 1\right), x, \varepsilon\right) \cdot x\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (* 0.3333333333333333 eps)
   eps
   (* (fma (fma 1.3333333333333333 (* (+ x eps) eps) 1.0) x eps) x))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma((0.3333333333333333 * eps), eps, (fma(fma(1.3333333333333333, ((x + eps) * eps), 1.0), x, eps) * x)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(Float64(0.3333333333333333 * eps), eps, Float64(fma(fma(1.3333333333333333, Float64(Float64(x + eps) * eps), 1.0), x, eps) * x)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * eps), $MachinePrecision] * eps + N[(N[(N[(1.3333333333333333 * N[(N[(x + eps), $MachinePrecision] * eps), $MachinePrecision] + 1.0), $MachinePrecision] * x + eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \left(\frac{4}{3} \cdot \left(\varepsilon \cdot x\right) + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \left(x + \varepsilon\right), 1\right), x, \varepsilon\right) \cdot x\right), \varepsilon, \varepsilon\right) \]

9. Final simplification 97.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \left(x + \varepsilon\right) \cdot \varepsilon, 1\right), x, \varepsilon\right) \cdot x\right), \varepsilon, \varepsilon\right) \]
10. Add Preprocessing

Alternative 10: 98.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma (fma 1.3333333333333333 (* eps eps) 1.0) x eps)
   x
   (* (* eps eps) 0.3333333333333333))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma(1.3333333333333333, (eps * eps), 1.0), x, eps), x, ((eps * eps) * 0.3333333333333333)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(1.3333333333333333, Float64(eps * eps), 1.0), x, eps), x, Float64(Float64(eps * eps) * 0.3333333333333333)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * x + eps), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 11: 98.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* (* eps eps) 0.3333333333333333) eps eps))

C

double code(double x, double eps) {
	return fma(((eps * eps) * 0.3333333333333333), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(eps * eps) * 0.3333333333333333), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

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 96.9%

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

Alternative 12: 98.0% 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 63.2%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.9%

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

Developer Target 1: 99.0% 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 2024273 
(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)))