2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.4%

Time: 12.7s

Alternatives: 10

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: 62.3% 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: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \tan x, 1\right)} \]
8. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.7%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 5: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.4%

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

Alternative 6: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return fma(fma(0.6666666666666666, (x * x), 1.0), (x * x), 1.0) * (fma(eps, tan(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) * Float64(fma(eps, tan(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] * N[(N[(eps * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.3%

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

Alternative 7: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

Alternative 8: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

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

Alternative 9: 97.9% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

Alternative 10: 6.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 6.6%

    \[\leadsto \left(x \cdot x\right) \cdot \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 6.6%

    \[\leadsto \left(\varepsilon \cdot x\right) \cdot x \]
14. 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 2024322 
(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)))