2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.7%

Time: 13.6s

Alternatives: 7

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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma -0.16666666666666666 (* eps eps) 1.0) eps)
  (* (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	return (fma(-0.16666666666666666, (eps * eps), 1.0) * eps) / (cos((eps + x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. lift--.f64 N/A

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

   2. 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-rev 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 62.2

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

4. Applied rewrites 62.2%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

10. Applied rewrites 99.2%

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

Alternative 3: 98.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (*
    (fma
     (fma
      (fma 0.19682539682539682 (* x x) 0.37777777777777777)
      (* x x)
      0.6666666666666666)
     (* x x)
     1.0)
    (* x x))
   1.0)
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(fma(0.19682539682539682, (x * x), 0.37777777777777777), (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(fma(0.19682539682539682, Float64(x * x), 0.37777777777777777), Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(0.19682539682539682 * N[(x * x), $MachinePrecision] + 0.37777777777777777), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 4: 98.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right), x \cdot x, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0)
   (* x x)
   1.0)
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(0.37777777777777777, Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0), Float64(x * x), 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.37777777777777777 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

    \[\leadsto \left(x \cdot x + 1\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto \left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

14. Applied rewrites 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right), x \cdot x, 1\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 5: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 1\right), x \cdot x, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (fma 0.6666666666666666 (* x x) 1.0) (* x x) 1.0) eps))

C

double code(double x, double eps) {
	return fma(fma(0.6666666666666666, (x * x), 1.0), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(0.6666666666666666, Float64(x * x), 1.0), Float64(x * x), 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

    \[\leadsto \left(x \cdot x + 1\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 1\right), x \cdot x, 1\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 6: 98.2% 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 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

    \[\leadsto \left(x \cdot x + 1\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 98.5%

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

Alternative 7: 97.8% 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 62.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-1 \cdot \varepsilon\right) \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right), \sin x, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \sin x\right) \cdot -0.3333333333333333\right)}{\cos x} - \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1 \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, -1\right), {\sin x}^{2}, {\sin x}^{2} \cdot 0.16666666666666666\right)}{{\cos x}^{2}} + \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right) + 0.16666666666666666\right)\right), \varepsilon, \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right) \cdot \frac{\sin x}{\cos x}\right) \cdot 1\right), \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 1, 1\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

    \[\leadsto \left(x \cdot x + 1\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 97.9%

    \[\leadsto 1 \cdot \varepsilon \]
15. 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 2024313 
(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)))