2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.9%

Time: 13.3s

Alternatives: 6

Speedup: 34.5×

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: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.0

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

4. Applied rewrites 59.0%

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

5. Taylor expanded in eps around -inf

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

   1. mul-1-neg N/A

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

   2. remove-double-neg N/A

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

   3. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 98.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

Alternative 3: 98.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.9%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 98.9%

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

Alternative 4: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.8%

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

Alternative 5: 98.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.8%

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

Alternative 6: 97.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 1.0 eps))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0 * eps;
}

Python

def code(x, eps):
	return 1.0 * eps

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 * eps;
end

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in eps around 0

   \[\leadsto 1 \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.5%

    \[\leadsto 1 \cdot \varepsilon \]
12. 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 2024307 
(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)))