2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.6%

Time: 19.3s

Alternatives: 14

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: 62.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\tan x}^{2}\\ t_1 := \cos \left(x + x\right)\\ t_2 := t\_0 \cdot \left(\frac{\mathsf{fma}\left(t\_1, -0.5, 0.5\right)}{\mathsf{fma}\left(t\_1, 0.5, 0.5\right)} - -0.3333333333333333\right)\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\left(-\sin x\right)}^{2}, {\left(\frac{1}{\cos x}\right)}^{2}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, t\_2, t\_0 \cdot \frac{\sin x}{\cos x \cdot 3}\right), t\_2\right), \tan x \cdot t\_0\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow (tan x) 2.0)))
        (t_1 (cos (+ x x)))
        (t_2
         (*
          t_0
          (- (/ (fma t_1 -0.5 0.5) (fma t_1 0.5 0.5)) -0.3333333333333333))))
   (fma
    eps
    (fma
     (pow (- (sin x)) 2.0)
     (pow (/ 1.0 (cos x)) 2.0)
     (*
      eps
      (fma
       eps
       (fma eps (fma (tan x) t_2 (* t_0 (/ (sin x) (* (cos x) 3.0)))) t_2)
       (* (tan x) t_0))))
    eps)))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(tan(x), 2.0);
	double t_1 = cos((x + x));
	double t_2 = t_0 * ((fma(t_1, -0.5, 0.5) / fma(t_1, 0.5, 0.5)) - -0.3333333333333333);
	return fma(eps, fma(pow(-sin(x), 2.0), pow((1.0 / cos(x)), 2.0), (eps * fma(eps, fma(eps, fma(tan(x), t_2, (t_0 * (sin(x) / (cos(x) * 3.0)))), t_2), (tan(x) * t_0)))), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (tan(x) ^ 2.0))
	t_1 = cos(Float64(x + x))
	t_2 = Float64(t_0 * Float64(Float64(fma(t_1, -0.5, 0.5) / fma(t_1, 0.5, 0.5)) - -0.3333333333333333))
	return fma(eps, fma((Float64(-sin(x)) ^ 2.0), (Float64(1.0 / cos(x)) ^ 2.0), Float64(eps * fma(eps, fma(eps, fma(tan(x), t_2, Float64(t_0 * Float64(sin(x) / Float64(cos(x) * 3.0)))), t_2), Float64(tan(x) * t_0)))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] / N[(t$95$1 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[Power[(-N[Sin[x], $MachinePrecision]), 2.0], $MachinePrecision] * N[Power[N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(eps * N[(eps * N[(eps * N[(N[Tan[x], $MachinePrecision] * t$95$2 + N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]]]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\left(-\sin x\right)}^{2}, {\left(\frac{1}{\cos x}\right)}^{2}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \left({\tan x}^{2} + 1\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right), \left({\tan x}^{2} + 1\right) \cdot \frac{\sin x}{\cos x \cdot 3}\right), \left({\tan x}^{2} + 1\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right)\right), \left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot \varepsilon\right), \varepsilon\right) \]

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\left(-\sin x\right)}^{2}, {\left(\frac{1}{\cos x}\right)}^{2}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \left(1 + {\tan x}^{2}\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right), \left(1 + {\tan x}^{2}\right) \cdot \frac{\sin x}{\cos x \cdot 3}\right), \left(1 + {\tan x}^{2}\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right)\right), \tan x \cdot \left(1 + {\tan x}^{2}\right)\right)\right), \varepsilon\right) \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := {\tan x}^{2}\\ t_2 := 1 + t\_1\\ t_3 := t\_2 \cdot \left(\frac{\mathsf{fma}\left(t\_0, -0.5, 0.5\right)}{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)} - -0.3333333333333333\right)\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, t\_3, t\_2 \cdot \frac{\sin x}{\cos x \cdot 3}\right), t\_3\right), \tan x \cdot t\_2\right), \varepsilon, t\_1\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (cos (+ x x)))
        (t_1 (pow (tan x) 2.0))
        (t_2 (+ 1.0 t_1))
        (t_3
         (*
          t_2
          (- (/ (fma t_0 -0.5 0.5) (fma t_0 0.5 0.5)) -0.3333333333333333))))
   (fma
    eps
    (fma
     (fma
      eps
      (fma eps (fma (tan x) t_3 (* t_2 (/ (sin x) (* (cos x) 3.0)))) t_3)
      (* (tan x) t_2))
     eps
     t_1)
    eps)))

C

double code(double x, double eps) {
	double t_0 = cos((x + x));
	double t_1 = pow(tan(x), 2.0);
	double t_2 = 1.0 + t_1;
	double t_3 = t_2 * ((fma(t_0, -0.5, 0.5) / fma(t_0, 0.5, 0.5)) - -0.3333333333333333);
	return fma(eps, fma(fma(eps, fma(eps, fma(tan(x), t_3, (t_2 * (sin(x) / (cos(x) * 3.0)))), t_3), (tan(x) * t_2)), eps, t_1), eps);
}

Julia

function code(x, eps)
	t_0 = cos(Float64(x + x))
	t_1 = tan(x) ^ 2.0
	t_2 = Float64(1.0 + t_1)
	t_3 = Float64(t_2 * Float64(Float64(fma(t_0, -0.5, 0.5) / fma(t_0, 0.5, 0.5)) - -0.3333333333333333))
	return fma(eps, fma(fma(eps, fma(eps, fma(tan(x), t_3, Float64(t_2 * Float64(sin(x) / Float64(cos(x) * 3.0)))), t_3), Float64(tan(x) * t_2)), eps, t_1), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] / N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(eps * N[(eps * N[(N[Tan[x], $MachinePrecision] * t$95$3 + N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * eps + t$95$1), $MachinePrecision] + eps), $MachinePrecision]]]]]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \left({\tan x}^{2} + 1\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right), \left({\tan x}^{2} + 1\right) \cdot \frac{\sin x}{\cos x \cdot 3}\right), \left({\tan x}^{2} + 1\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right)\right), \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \color{blue}{\varepsilon}, {\tan x}^{2}\right), \varepsilon\right) \]

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \left(1 + {\tan x}^{2}\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right), \left(1 + {\tan x}^{2}\right) \cdot \frac{\sin x}{\cos x \cdot 3}\right), \left(1 + {\tan x}^{2}\right) \cdot \left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - -0.3333333333333333\right)\right), \tan x \cdot \left(1 + {\tan x}^{2}\right)\right), \varepsilon, {\tan x}^{2}\right), \varepsilon\right) \]
13. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.8%

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

13. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left({\left(-\sin x\right)}^{2}, {\left(\frac{1}{\cos x}\right)}^{2}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \frac{\sin x \cdot \left(1 + {\tan x}^{2}\right)}{\cos x}\right)\right), \varepsilon\right) \]
14. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.8%

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

Alternative 5: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \sin x \cdot \frac{-1}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   (sin x)
   (* (sin x) (/ -1.0 (fma (cos (+ x x)) -0.5 -0.5)))
   (*
    eps
    (fma eps 0.3333333333333333 (fma (* eps eps) (* x 0.6666666666666666) x))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(sin(x), (sin(x) * (-1.0 / fma(cos((x + x)), -0.5, -0.5))), (eps * fma(eps, 0.3333333333333333, fma((eps * eps), (x * 0.6666666666666666), x)))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(sin(x), Float64(sin(x) * Float64(-1.0 / fma(cos(Float64(x + x)), -0.5, -0.5))), Float64(eps * fma(eps, 0.3333333333333333, fma(Float64(eps * eps), Float64(x * 0.6666666666666666), x)))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(-1.0 / N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * 0.3333333333333333 + N[(N[(eps * eps), $MachinePrecision] * N[(x * 0.6666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

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

10. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \color{blue}{\sin x \cdot \frac{-1}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)}}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666 \cdot x, x\right)\right)\right), \varepsilon\right) \]

11. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \sin x \cdot \frac{-1}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right)\right), \varepsilon\right) \]
12. Add Preprocessing

Alternative 6: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.4%

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

Alternative 7: 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 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 99.3%

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

14. Final simplification 99.3%

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

Alternative 8: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), {\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), 1\right)\right)}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   eps
   (fma eps 0.3333333333333333 (fma (* eps eps) (* x 0.6666666666666666) x))
   (pow
    (*
     x
     (fma
      (* x x)
      (fma
       (* x x)
       (fma (* x x) 0.05396825396825397 0.13333333333333333)
       0.3333333333333333)
      1.0))
    2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma((eps * eps), (x * 0.6666666666666666), x)), pow((x * fma((x * x), fma((x * x), fma((x * x), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), 1.0)), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma(Float64(eps * eps), Float64(x * 0.6666666666666666), x)), (Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), 1.0)) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333 + N[(N[(eps * eps), $MachinePrecision] * N[(x * 0.6666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \frac{1}{3}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{2}{3} \cdot x, x\right)\right), {\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {x}^{2}\right)\right)\right)\right)}^{2}\right), \varepsilon, \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666 \cdot x, x\right)\right), {\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), 1\right)\right)}^{2}\right), \varepsilon, \varepsilon\right) \]

14. Final simplification 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), {\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), 1\right)\right)}^{2}\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 9: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   eps
   (fma eps 0.3333333333333333 (fma (* eps eps) (* x 0.6666666666666666) x))
   (*
    (* x x)
    (fma
     (* x x)
     (fma
      (* x x)
      (fma (* x x) 0.19682539682539682 0.37777777777777777)
      0.6666666666666666)
     1.0)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma((eps * eps), (x * 0.6666666666666666), x)), ((x * x) * fma((x * x), fma((x * x), fma((x * x), 0.19682539682539682, 0.37777777777777777), 0.6666666666666666), 1.0))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma(Float64(eps * eps), Float64(x * 0.6666666666666666), x)), Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.19682539682539682, 0.37777777777777777), 0.6666666666666666), 1.0))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333 + N[(N[(eps * eps), $MachinePrecision] * N[(x * 0.6666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.19682539682539682 + 0.37777777777777777), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666 \cdot x, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

14. Final simplification 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 10: 98.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   eps
   (fma eps 0.3333333333333333 (fma (* eps eps) (* x 0.6666666666666666) x))
   (*
    (* x x)
    (fma (* x x) (fma (* x x) 0.37777777777777777 0.6666666666666666) 1.0)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma((eps * eps), (x * 0.6666666666666666), x)), ((x * x) * fma((x * x), fma((x * x), 0.37777777777777777, 0.6666666666666666), 1.0))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, fma(Float64(eps * eps), Float64(x * 0.6666666666666666), x)), Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), 0.37777777777777777, 0.6666666666666666), 1.0))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(eps * 0.3333333333333333 + N[(N[(eps * eps), $MachinePrecision] * N[(x * 0.6666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.37777777777777777 + 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666 \cdot x, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]

14. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot 0.6666666666666666, x\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 11: 98.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.8%

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

14. Final simplification 98.8%

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

Alternative 12: 98.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)}, 0.3333333333333333 \cdot \varepsilon\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.6%

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

14. Final simplification 98.6%

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

Alternative 13: 97.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 97.9%

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

Alternative 14: 97.9% accurate, 17.3× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 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

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 97.9%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 97.9%

    \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \varepsilon, \varepsilon\right) \]
15. 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: 99.0% 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 2024222 
(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)))