2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 100.0%

Time: 19.2s

Alternatives: 17

Speedup: 12.2×

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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(0 - \sin \varepsilon\right)\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (* (cos x) (fma (cos x) (cos eps) (* (sin x) (- 0.0 (sin eps)))))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * fma(cos(x), cos(eps), (sin(x) * (0.0 - sin(eps)))));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * fma(cos(x), cos(eps), Float64(sin(x) * Float64(0.0 - sin(eps))))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(0.0 - N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.8%

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

   1. cos-sum N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. cos-lowering-cos.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. sin-lowering-sin.f64 100.0

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \frac{\sin \varepsilon}{\cos x \cdot \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(0 - \sin \varepsilon\right)\right)} \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\varepsilon + x\right) - \tan x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right), x\right)\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan (+ eps x)) (tan x))))
   (if (<= t_0 2e-11)
     (fma
      eps
      (fma
       0.3333333333333333
       (* eps eps)
       (* x (fma eps (fma 0.6666666666666666 (* eps eps) 1.0) x)))
      eps)
     t_0)))

C

double code(double x, double eps) {
	double t_0 = tan((eps + x)) - tan(x);
	double tmp;
	if (t_0 <= 2e-11) {
		tmp = fma(eps, fma(0.3333333333333333, (eps * eps), (x * fma(eps, fma(0.6666666666666666, (eps * eps), 1.0), x))), eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(Float64(eps + x)) - tan(x))
	tmp = 0.0
	if (t_0 <= 2e-11)
		tmp = fma(eps, fma(0.3333333333333333, Float64(eps * eps), Float64(x * fma(eps, fma(0.6666666666666666, Float64(eps * eps), 1.0), x))), eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-11], N[(eps * N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision] + N[(x * N[(eps * N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x)) < 1.99999999999999988e-11

   1. Initial program 60.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

   8. Simplified 100.0%

      \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

   9. Taylor expanded in x around 0

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

       1. accelerator-lowering-fma.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. +-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\frac{2}{3}, {\varepsilon}^{2}, 1\right)}, x\right)\right), \varepsilon\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{2}{3}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right), x\right)\right), \varepsilon\right) \]

       10. *-lowering-*.f64 100.0

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

   11. Simplified 100.0%

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

   if 1.99999999999999988e-11 < (-.f64 (tan.f64 (+.f64 x eps)) (tan.f64 x))

   1. Initial program 76.6%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan \left(\varepsilon + x\right) - \tan x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right), x\right)\right), \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (fma
    (* eps eps)
    (fma
     (* eps eps)
     (fma (* eps eps) -0.0001984126984126984 0.008333333333333333)
     -0.16666666666666666)
    1.0))
  (* (cos x) (cos (+ eps x)))))

C

double code(double x, double eps) {
	return (eps * fma((eps * eps), fma((eps * eps), fma((eps * eps), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)) / (cos(x) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * fma(Float64(eps * eps), fma(Float64(eps * eps), fma(Float64(eps * eps), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)) / Float64(cos(x) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, {\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}, 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, {\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, {\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. sub-neg N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right)}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   11. +-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{-1}{5040} \cdot {\varepsilon}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   15. *-lowering-*.f64 99.8

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

7. Simplified 99.8%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

8. Final simplification 99.8%

   \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
9. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   eps
   (fma
    eps
    (* eps (fma (* eps eps) 0.008333333333333333 -0.16666666666666666))
    1.0))
  (* (cos x) (cos (+ eps x)))))

C

double code(double x, double eps) {
	return (eps * fma(eps, (eps * fma((eps * eps), 0.008333333333333333, -0.16666666666666666)), 1.0)) / (cos(x) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * fma(eps, Float64(eps * fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666)), 1.0)) / Float64(cos(x) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{\varepsilon \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)} + 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right), 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   7. sub-neg N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left({\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

   12. *-lowering-*.f64 99.7

       \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

7. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

8. Final simplification 99.7%

   \[\leadsto \frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
9. Add Preprocessing

Alternative 6: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 99.4

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 7: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

   \[\leadsto \frac{\varepsilon}{\cos x \cdot \cos \left(\varepsilon + x\right)} \]
9. Add Preprocessing

Alternative 8: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (cos(x) ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps / Math.pow(Math.cos(x), 2.0);
}

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 61.2%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sin-diff N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 N/A

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

   13. +-lowering-+.f64 61.1

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

4. Applied egg-rr 61.1%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\varepsilon}{\color{blue}{{\cos x}^{2}}} \]

   3. cos-lowering-cos.f64 97.7

      \[\leadsto \frac{\varepsilon}{{\color{blue}{\cos x}}^{2}} \]

7. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 9: 98.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Simplified 97.2%

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

Alternative 10: 98.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

    1. accelerator-lowering-fma.f64 N/A

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 N/A

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

    4. *-lowering-*.f64 N/A

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

    5. +-commutative N/A

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

    6. accelerator-lowering-fma.f64 N/A

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

    7. +-commutative N/A

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

    8. *-commutative N/A

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

    9. unpow2 N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. accelerator-lowering-fma.f64 N/A

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

    13. *-commutative N/A

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

    14. *-lowering-*.f64 N/A

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

    15. *-lowering-*.f64 N/A

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

    16. +-commutative N/A

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

    17. accelerator-lowering-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{2}{3}, 1\right), \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3}, {\varepsilon}^{2}, 1\right)}\right)\right), \varepsilon\right) \]

    18. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{2}{3}, 1\right), \varepsilon \cdot \mathsf{fma}\left(\frac{2}{3}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right)\right)\right), \varepsilon\right) \]

    19. *-lowering-*.f64 97.2

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

11. Simplified 97.2%

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

Alternative 11: 98.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Simplified 97.2%

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

12. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 N/A

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

    2. +-commutative N/A

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

    3. accelerator-lowering-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3} + 1, \varepsilon\right), \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)} + 1, \varepsilon\right), \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{2}{3} \cdot x\right)} + 1, \varepsilon\right), \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    9. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} \cdot x, 1\right)}, \varepsilon\right), \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}}, 1\right), \varepsilon\right), \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    11. *-lowering-*.f64 97.2

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

14. Simplified 97.2%

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

Alternative 12: 98.5% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Simplified 97.2%

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

12. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

    4. +-commutative N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. accelerator-lowering-fma.f64 N/A

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

    9. +-commutative N/A

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

    10. unpow2 N/A

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

    11. associate-*l* N/A

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

    12. accelerator-lowering-fma.f64 N/A

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

14. Simplified 97.2%

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

Alternative 13: 98.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Simplified 97.2%

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

12. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

    4. +-commutative N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*l* N/A

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

    9. accelerator-lowering-fma.f64 N/A

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

    10. *-lowering-*.f64 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3} + 1\right), 1\right)\right) \]

    14. associate-*l* N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)} + 1\right), 1\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x, \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{2}{3} \cdot x\right)} + 1\right), 1\right)\right) \]

    16. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} \cdot x, 1\right)}, 1\right)\right) \]

    17. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}}, 1\right), 1\right)\right) \]

    18. *-lowering-*.f64 97.0

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

14. Simplified 97.0%

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

Alternative 14: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Simplified 97.2%

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

12. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 N/A

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

    2. +-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. +-commutative N/A

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

    8. *-commutative N/A

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

    9. unpow2 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3} + 1\right), 1\right) \]

    10. associate-*l* N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)} + 1\right), 1\right) \]

    11. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{2}{3} \cdot x\right)} + 1\right), 1\right) \]

    12. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} \cdot x, 1\right)}, 1\right) \]

    13. *-commutative N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}}, 1\right), 1\right) \]

    14. *-lowering-*.f64 96.8

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

14. Simplified 96.8%

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

Alternative 15: 98.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{1}{3} \cdot {\varepsilon}^{2}}, \varepsilon\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon\right) \]

   3. *-lowering-*.f64 96.5

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

8. Simplified 96.5%

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

9. Final simplification 96.5%

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

Alternative 16: 98.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), \color{blue}{0.3333333333333333 \cdot \varepsilon}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

8. Simplified 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right), 0.3333333333333333 \cdot \varepsilon\right)}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon + \frac{1}{3} \cdot {\varepsilon}^{3}} \]
10. Step-by-step derivation

    1. *-rgt-identity N/A

       \[\leadsto \color{blue}{\varepsilon \cdot 1} + \frac{1}{3} \cdot {\varepsilon}^{3} \]

    2. cube-mult N/A

       \[\leadsto \varepsilon \cdot 1 + \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. distribute-lft-in N/A

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

    8. *-lowering-*.f64 N/A

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

    9. +-commutative N/A

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

    10. accelerator-lowering-fma.f64 N/A

        \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{2}, 1\right)} \]

    11. unpow2 N/A

        \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \]

    12. *-lowering-*.f64 96.5

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

11. Simplified 96.5%

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

Alternative 17: 7.9% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ 0 - x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 x))

C

double code(double x, double eps) {
	return 0.0 - x;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - x
end function

Java

public static double code(double x, double eps) {
	return 0.0 - x;
}

Python

def code(x, eps):
	return 0.0 - x

Julia

function code(x, eps)
	return Float64(0.0 - x)
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - x;
end

Wolfram

code[x_, eps_] := N[(0.0 - x), $MachinePrecision]

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 58.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot x} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - x} \]

   3. --lowering--.f64 8.0

      \[\leadsto \color{blue}{0 - x} \]

8. Simplified 8.0%

   \[\leadsto \color{blue}{0 - x} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 8.0

      \[\leadsto \color{blue}{-x} \]

10. Applied egg-rr 8.0%

    \[\leadsto \color{blue}{-x} \]

11. Final simplification 8.0%

    \[\leadsto 0 - x \]
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: 62.6% 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: 99.1% 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 2024194 
(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)))