2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.6%

Time: 15.8s

Alternatives: 12

Speedup: 17.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := 0.5 \cdot \cos \left(x + x\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{\left(0.5 - t\_1\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{0.5 + t\_1} - \mathsf{fma}\left(t\_0, -0.3333333333333333, -0.3333333333333333\right), \frac{\mathsf{fma}\left(\sin x, t\_0, \sin x\right)}{\cos x}\right), t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (* 0.5 (cos (+ x x)))))
   (fma
    (fma
     eps
     (fma
      eps
      (-
       (/ (+ (- 0.5 t_1) (pow (* (sin x) (tan x)) 2.0)) (+ 0.5 t_1))
       (fma t_0 -0.3333333333333333 -0.3333333333333333))
      (/ (fma (sin x) t_0 (sin x)) (cos x)))
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = 0.5 * cos((x + x));
	return fma(fma(eps, fma(eps, ((((0.5 - t_1) + pow((sin(x) * tan(x)), 2.0)) / (0.5 + t_1)) - fma(t_0, -0.3333333333333333, -0.3333333333333333)), (fma(sin(x), t_0, sin(x)) / cos(x))), t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(0.5 * cos(Float64(x + x)))
	return fma(fma(eps, fma(eps, Float64(Float64(Float64(Float64(0.5 - t_1) + (Float64(sin(x) * tan(x)) ^ 2.0)) / Float64(0.5 + t_1)) - fma(t_0, -0.3333333333333333, -0.3333333333333333)), Float64(fma(sin(x), t_0, sin(x)) / cos(x))), t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(eps * N[(N[(N[(N[(0.5 - t$95$1), $MachinePrecision] + N[Power[N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * -0.3333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$0 + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{0.5 + 0.5 \cdot \cos \left(x + x\right)} - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.3333333333333333\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right), {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \tan x + {\tan x}^{3}, {\tan x}^{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* eps (fma eps (+ (tan x) (pow (tan x) 3.0)) (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	return eps + (eps * fma(eps, (tan(x) + pow(tan(x), 3.0)), pow(tan(x), 2.0)));
}

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * fma(eps, Float64(tan(x) + (tan(x) ^ 3.0)), (tan(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(eps * N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 99.2

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

10. Simplified 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \varepsilon + \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \tan x + {\tan x}^{3}, {\tan x}^{2}\right) \]
14. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x + {\tan x}^{3}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma (+ (tan x) (pow (tan x) 3.0)) eps (pow (tan x) 2.0)) eps eps))

C

double code(double x, double eps) {
	return fma(fma((tan(x) + pow(tan(x), 3.0)), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(Float64(tan(x) + (tan(x) ^ 3.0)), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 99.2

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

10. Simplified 99.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-cos.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-sin.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. lift-pow.f64 N/A

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

    8. lift-+.f64 N/A

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

    9. lift-tan.f64 N/A

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

    10. lift-pow.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f64 99.2

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x + {\tan x}^{3}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
14. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), {\left(\frac{1}{\tan x}\right)}^{-2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma eps (fma eps 0.3333333333333333 x) (pow (/ 1.0 (tan x)) -2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), pow((1.0 / tan(x)), -2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), (Float64(1.0 / tan(x)) ^ -2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333 + x), $MachinePrecision] + N[Power[N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Simplified 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

    1. tan-quot N/A

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

    2. lift-sin.f64 N/A

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

    3. lift-cos.f64 N/A

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

    4. clear-num N/A

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

    5. inv-pow N/A

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

    6. pow-pow N/A

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

    7. lower-pow.f64 N/A

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

    8. clear-num N/A

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

    9. lift-sin.f64 N/A

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

    10. lift-cos.f64 N/A

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

    11. tan-quot N/A

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

    12. lift-tan.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. metadata-eval 98.6

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), {\left(\frac{1}{\tan x}\right)}^{\color{blue}{-2}}\right), \varepsilon, \varepsilon\right) \]

12. Applied egg-rr 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), \color{blue}{{\left(\frac{1}{\tan x}\right)}^{-2}}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 5: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right), {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma eps (fma eps 0.3333333333333333 x) (pow (tan x) 2.0)) eps eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333 + x), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Simplified 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 6: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.3333333333333333, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma eps (* eps 0.3333333333333333) (pow (tan x) 2.0)) eps eps))

C

double code(double x, double eps) {
	return fma(fma(eps, (eps * 0.3333333333333333), pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, Float64(eps * 0.3333333333333333), (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 98.5

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.3333333333333333}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Simplified 98.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot 0.3333333333333333}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 7: 98.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Simplified 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Taylor expanded in x around 0

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

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. associate-+r+ N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. distribute-lft-in N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. +-commutative N/A

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

    16. distribute-rgt-in N/A

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

13. Simplified 97.9%

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

Alternative 8: 98.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 99.2

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

10. Simplified 99.2%

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

11. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon\right)\right)\right)}, \varepsilon, \varepsilon\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon\right)\right) + \varepsilon\right)}, \varepsilon, \varepsilon\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon\right), \varepsilon\right)}, \varepsilon, \varepsilon\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon\right) + 1}, \varepsilon\right), \varepsilon, \varepsilon\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} \cdot x + \frac{4}{3} \cdot \varepsilon, 1\right)}, \varepsilon\right), \varepsilon, \varepsilon\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{4}{3} \cdot \varepsilon + \frac{2}{3} \cdot x}, 1\right), \varepsilon\right), \varepsilon, \varepsilon\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{4}{3}} + \frac{2}{3} \cdot x, 1\right), \varepsilon\right), \varepsilon, \varepsilon\right) \]

    8. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{4}{3}, \frac{2}{3} \cdot x\right)}, 1\right), \varepsilon\right), \varepsilon, \varepsilon\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{4}{3}, \color{blue}{x \cdot \frac{2}{3}}\right), 1\right), \varepsilon\right), \varepsilon, \varepsilon\right) \]

    10. lower-*.f64 97.9

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

13. Simplified 97.9%

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

Alternative 9: 98.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Simplified 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Taylor expanded in x around 0

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

    1. distribute-rgt-in N/A

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

    2. unpow2 N/A

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

    3. associate-+r+ N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. distribute-lft-in N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. lower-fma.f64 N/A

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

    14. unpow2 N/A

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

    15. lower-*.f64 97.8

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

13. Simplified 97.8%

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

Alternative 10: 98.5% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 99.2

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

10. Simplified 99.2%

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

11. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 97.8

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

13. Simplified 97.8%

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

Alternative 11: 98.0% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 99.2

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

10. Simplified 99.2%

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

11. Taylor expanded in x around 0

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

    1. lower-*.f64 97.0

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

13. Simplified 97.0%

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

Alternative 12: 5.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 97.0

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

10. Simplified 97.0%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 5.8

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

13. Simplified 5.8%

    \[\leadsto \color{blue}{x \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
14. 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: 62.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(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)))