2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.7%

Time: 15.4s

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: 62.5% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \tan x\\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right), \cos x \cdot \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right), \varepsilon, \varepsilon \cdot t\_0\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (tan x))))
   (/
    (fma
     (fma
      t_0
      (* 0.3333333333333333 (* eps eps))
      (* (cos x) (fma 0.3333333333333333 (* eps eps) 1.0)))
     eps
     (* eps t_0))
    (* (cos x) (- 1.0 (* (tan x) (tan eps)))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * tan(x);
	return fma(fma(t_0, (0.3333333333333333 * (eps * eps)), (cos(x) * fma(0.3333333333333333, (eps * eps), 1.0))), eps, (eps * t_0)) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * tan(x))
	return Float64(fma(fma(t_0, Float64(0.3333333333333333 * Float64(eps * eps)), Float64(cos(x) * fma(0.3333333333333333, Float64(eps * eps), 1.0))), eps, Float64(eps * t_0)) / Float64(cos(x) * Float64(1.0 - Float64(tan(x) * tan(eps)))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. tan-sum N/A

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

   10. frac-add N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \sin x + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{\cos x} + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \cos x, \frac{{\sin x}^{2}}{\cos x} \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. tan-sum N/A

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

   10. frac-add N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \sin x + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{\cos x} + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \cos x, \frac{{\sin x}^{2}}{\cos x} \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. tan-sum N/A

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

   10. frac-add N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \sin x + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{\cos x} + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \cos x, \frac{{\sin x}^{2}}{\cos x} \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. tan-sum N/A

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

   10. frac-add N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot \sin x + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0} + \varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right)\right) + 0}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \cos x + \left({\varepsilon}^{2} \cdot \left(\frac{1}{3} \cdot \cos x + \frac{1}{3} \cdot \frac{{\sin x}^{2}}{\cos x}\right) + \frac{{\sin x}^{2}}{\cos x}\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{\cos x} + \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \cos x, \frac{{\sin x}^{2}}{\cos x} \cdot \left(0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), 0\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

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

Alternative 5: 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 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.5%

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

Alternative 6: 98.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666, \varepsilon\right)\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   0.3333333333333333
   (* eps eps)
   (*
    x
    (fma
     x
     (fma 1.3333333333333333 (* eps eps) 1.0)
     (fma eps (* (* eps eps) 0.6666666666666666) eps))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(0.3333333333333333, (eps * eps), (x * fma(x, fma(1.3333333333333333, (eps * eps), 1.0), fma(eps, ((eps * eps) * 0.6666666666666666), eps)))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(0.3333333333333333, Float64(eps * eps), Float64(x * fma(x, fma(1.3333333333333333, Float64(eps * eps), 1.0), fma(eps, Float64(Float64(eps * eps) * 0.6666666666666666), eps)))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision] + N[(x * N[(x * N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] + N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}, \left(\left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right) + -0.16666666666666666\right) \cdot \frac{\sin x}{\cos x}\right), \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right) + -0.16666666666666666\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \color{blue}{\varepsilon \cdot \varepsilon}, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \mathsf{fma}\left(\varepsilon, 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)\right)\right), \varepsilon\right) \]

9. Final simplification 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), \mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.6666666666666666, \varepsilon\right)\right)\right), \varepsilon\right) \]
10. Add Preprocessing

Alternative 7: 98.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

Alternative 8: 98.4% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

Alternative 9: 98.3% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

Alternative 10: 98.2% accurate, 17.3× speedup

Math

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

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(x, Float64(x * eps), eps)
end

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

Alternative 11: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 12: 6.4% accurate, 18.8× speedup

Math

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

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(eps * Float64(x * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 6.4%

    \[\leadsto \varepsilon \cdot \left(x \cdot \color{blue}{x}\right) \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)

Julia

function code(x, eps)
	return Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
end

Wolfram

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

Developer Target 3: 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 2024232 
(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 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

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

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