2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.6%

Time: 17.3s

Alternatives: 15

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: 62.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := \frac{{\sin x}^{2}}{t\_0}\\ t_3 := \mathsf{fma}\left(1.3333333333333333, t\_2, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + t\_3, t\_1, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{t\_0}\right)}{\cos x}\right), t\_3\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(t\_2, \frac{-1}{\varepsilon} - t\_1, -t\_1\right) \cdot \left(-\varepsilon\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (/ (pow (sin x) 2.0) t_0))
        (t_3
         (fma 1.3333333333333333 t_2 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
   (fma
    eps
    (fma
     (+
      0.3333333333333333
      (fma
       eps
       (fma
        (+ 0.3333333333333333 t_3)
        t_1
        (/
         (* 0.3333333333333333 (+ (sin x) (/ (pow (sin x) 3.0) t_0)))
         (cos x)))
       t_3))
     (* eps eps)
     (* (fma t_2 (- (/ -1.0 eps) t_1) (- t_1)) (- eps)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = sin(x) / cos(x);
	double t_2 = pow(sin(x), 2.0) / t_0;
	double t_3 = fma(1.3333333333333333, t_2, (pow(sin(x), 4.0) / pow(cos(x), 4.0)));
	return fma(eps, fma((0.3333333333333333 + fma(eps, fma((0.3333333333333333 + t_3), t_1, ((0.3333333333333333 * (sin(x) + (pow(sin(x), 3.0) / t_0))) / cos(x))), t_3)), (eps * eps), (fma(t_2, ((-1.0 / eps) - t_1), -t_1) * -eps)), eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64(sin(x) / cos(x))
	t_2 = Float64((sin(x) ^ 2.0) / t_0)
	t_3 = fma(1.3333333333333333, t_2, Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))
	return fma(eps, fma(Float64(0.3333333333333333 + fma(eps, fma(Float64(0.3333333333333333 + t_3), t_1, Float64(Float64(0.3333333333333333 * Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_0))) / cos(x))), t_3)), Float64(eps * eps), Float64(fma(t_2, Float64(Float64(-1.0 / eps) - t_1), Float64(-t_1)) * Float64(-eps))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(1.3333333333333333 * t$95$2 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(0.3333333333333333 + N[(eps * N[(N[(0.3333333333333333 + t$95$3), $MachinePrecision] * t$95$1 + N[(N[(0.3333333333333333 * N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(N[(t$95$2 * N[(N[(-1.0 / eps), $MachinePrecision] - t$95$1), $MachinePrecision] + (-t$95$1)), $MachinePrecision] * (-eps)), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]]]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

11. Simplified 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)}, \varepsilon\right) \]

12. Taylor expanded in eps around -inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

14. Simplified 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{\varepsilon} - \frac{\sin x}{\cos x}, \frac{\sin x}{-\cos x}\right) \cdot \left(-\varepsilon\right)}\right), \varepsilon\right) \]

15. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-1}{\varepsilon} - \frac{\sin x}{\cos x}, -\frac{\sin x}{\cos x}\right) \cdot \left(-\varepsilon\right)\right), \varepsilon\right) \]
16. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{t\_0}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + t\_2, t\_1, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{t\_0}\right)}{\cos x}\right), t\_2\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(t\_1, \varepsilon + t\_1, \varepsilon \cdot {t\_1}^{3}\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (/ (sin x) (cos x)))
        (t_2
         (fma
          1.3333333333333333
          (/ (pow (sin x) 2.0) t_0)
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
   (fma
    eps
    (fma
     (+
      0.3333333333333333
      (fma
       eps
       (fma
        (+ 0.3333333333333333 t_2)
        t_1
        (/
         (* 0.3333333333333333 (+ (sin x) (/ (pow (sin x) 3.0) t_0)))
         (cos x)))
       t_2))
     (* eps eps)
     (fma t_1 (+ eps t_1) (* eps (pow t_1 3.0))))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = sin(x) / cos(x);
	double t_2 = fma(1.3333333333333333, (pow(sin(x), 2.0) / t_0), (pow(sin(x), 4.0) / pow(cos(x), 4.0)));
	return fma(eps, fma((0.3333333333333333 + fma(eps, fma((0.3333333333333333 + t_2), t_1, ((0.3333333333333333 * (sin(x) + (pow(sin(x), 3.0) / t_0))) / cos(x))), t_2)), (eps * eps), fma(t_1, (eps + t_1), (eps * pow(t_1, 3.0)))), eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64(sin(x) / cos(x))
	t_2 = fma(1.3333333333333333, Float64((sin(x) ^ 2.0) / t_0), Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))
	return fma(eps, fma(Float64(0.3333333333333333 + fma(eps, fma(Float64(0.3333333333333333 + t_2), t_1, Float64(Float64(0.3333333333333333 * Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_0))) / cos(x))), t_2)), Float64(eps * eps), fma(t_1, Float64(eps + t_1), Float64(eps * (t_1 ^ 3.0)))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(0.3333333333333333 + N[(eps * N[(N[(0.3333333333333333 + t$95$2), $MachinePrecision] * t$95$1 + N[(N[(0.3333333333333333 * N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(t$95$1 * N[(eps + t$95$1), $MachinePrecision] + N[(eps * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

11. Simplified 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)}, \varepsilon\right) \]

12. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \varepsilon + \frac{\sin x}{\cos x}, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right), \varepsilon\right) \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, t\_1, t\_1 + \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), t\_0 + {t\_0}^{3}\right), t\_1\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (fma
    eps
    (fma
     eps
     (fma
      eps
      (fma
       0.3333333333333333
       t_1
       (+ t_1 (+ 0.3333333333333333 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
      (+ t_0 (pow t_0 3.0)))
     t_1)
    eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return fma(eps, fma(eps, fma(eps, fma(0.3333333333333333, t_1, (t_1 + (0.3333333333333333 + (pow(sin(x), 4.0) / pow(cos(x), 4.0))))), (t_0 + pow(t_0, 3.0))), t_1), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return fma(eps, fma(eps, fma(eps, fma(0.3333333333333333, t_1, Float64(t_1 + Float64(0.3333333333333333 + Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))))), Float64(t_0 + (t_0 ^ 3.0))), t_1), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(eps * N[(eps * N[(0.3333333333333333 * t$95$1 + N[(t$95$1 + N[(0.3333333333333333 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + eps), $MachinePrecision]]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \mathsf{fma}\left(t\_0, \varepsilon + t\_0, \varepsilon \cdot {t\_0}^{3}\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (fma
    eps
    (fma
     eps
     (*
      eps
      (+
       0.3333333333333333
       (fma
        1.3333333333333333
        (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
        (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
     (fma t_0 (+ eps t_0) (* eps (pow t_0 3.0))))
    eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return fma(eps, fma(eps, (eps * (0.3333333333333333 + fma(1.3333333333333333, (pow(sin(x), 2.0) / pow(cos(x), 2.0)), (pow(sin(x), 4.0) / pow(cos(x), 4.0))))), fma(t_0, (eps + t_0), (eps * pow(t_0, 3.0)))), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return fma(eps, fma(eps, Float64(eps * Float64(0.3333333333333333 + fma(1.3333333333333333, Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)), Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))))), fma(t_0, Float64(eps + t_0), Float64(eps * (t_0 ^ 3.0)))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(eps * N[(eps * N[(0.3333333333333333 + N[(1.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(eps + t$95$0), $MachinePrecision] + N[(eps * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

11. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)}, \varepsilon\right) \]

12. Final simplification 99.5%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \mathsf{fma}\left(\frac{\sin x}{\cos x}, \varepsilon + \frac{\sin x}{\cos x}, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right), \varepsilon\right) \]
13. Add Preprocessing

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(t\_0, \varepsilon + t\_0, \varepsilon \cdot {t\_0}^{3}\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (fma
    eps
    (fma
     (+ 0.3333333333333333 (* x (* eps 0.6666666666666666)))
     (* eps eps)
     (fma t_0 (+ eps t_0) (* eps (pow t_0 3.0))))
    eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return fma(eps, fma((0.3333333333333333 + (x * (eps * 0.6666666666666666))), (eps * eps), fma(t_0, (eps + t_0), (eps * pow(t_0, 3.0)))), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return fma(eps, fma(Float64(0.3333333333333333 + Float64(x * Float64(eps * 0.6666666666666666))), Float64(eps * eps), fma(t_0, Float64(eps + t_0), Float64(eps * (t_0 ^ 3.0)))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(0.3333333333333333 + N[(x * N[(eps * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(t$95$0 * N[(eps + t$95$0), $MachinePrecision] + N[(eps * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

11. Simplified 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \frac{\sin x}{\cos x}, \frac{0.3333333333333333 \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)}{\cos x}\right), \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)}, \varepsilon\right) \]

12. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 99.0

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + x \cdot \color{blue}{\left(\varepsilon \cdot 0.6666666666666666\right)}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right), \varepsilon\right) \]

14. Simplified 99.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + \color{blue}{x \cdot \left(\varepsilon \cdot 0.6666666666666666\right)}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{\sin x}{\cos x} + \varepsilon, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right), \varepsilon\right) \]

15. Final simplification 99.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333 + x \cdot \left(\varepsilon \cdot 0.6666666666666666\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \varepsilon + \frac{\sin x}{\cos x}, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right), \varepsilon\right) \]
16. Add Preprocessing

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(t\_0, \varepsilon + t\_0, \varepsilon \cdot {t\_0}^{3}\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (fma eps (fma t_0 (+ eps t_0) (* eps (pow t_0 3.0))) eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return fma(eps, fma(t_0, (eps + t_0), (eps * pow(t_0, 3.0))), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return fma(eps, fma(t_0, Float64(eps + t_0), Float64(eps * (t_0 ^ 3.0))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 * N[(eps + t$95$0), $MachinePrecision] + N[(eps * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-+r+ N/A

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

    4. unpow2 N/A

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

    5. unpow2 N/A

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

    6. times-frac N/A

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

    7. distribute-rgt-out N/A

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

    8. lower-fma.f64 N/A

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 7: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(t\_0, t\_0, \varepsilon \cdot {t\_0}^{3}\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (fma eps (fma t_0 t_0 (* eps (pow t_0 3.0))) eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return fma(eps, fma(t_0, t_0, (eps * pow(t_0, 3.0))), eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return fma(eps, fma(t_0, t_0, Float64(eps * (t_0 ^ 3.0))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 * t$95$0 + N[(eps * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\varepsilon \cdot \left(\frac{1}{3} \cdot \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x} + \frac{\sin x \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-+r+ N/A

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

    4. unpow2 N/A

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

    5. unpow2 N/A

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

    6. times-frac N/A

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

    7. distribute-rgt-out N/A

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

    8. lower-fma.f64 N/A

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

11. Simplified 98.9%

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

12. Taylor expanded in eps around 0

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

    1. lower-/.f64 N/A

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

    2. lower-sin.f64 N/A

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

    3. lower-cos.f64 98.4

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

14. Simplified 98.4%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{\sin x}{\cos x}, \color{blue}{\frac{\sin x}{\cos x}}, \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{3}\right), \varepsilon\right) \]
15. Add Preprocessing

Alternative 8: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

Alternative 9: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 10: 98.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 97.5

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

8. Simplified 97.5%

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

Alternative 11: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

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

   1. lower-fma.f64 N/A

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

8. Simplified 98.9%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 97.4

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 12: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Simplified 97.4%

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

Alternative 13: 97.8% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.8%

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

6. Taylor expanded in eps around 0

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

   1. lower-fma.f64 N/A

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

8. Simplified 98.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 97.2

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

11. Simplified 97.2%

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

12. Final simplification 97.2%

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

Alternative 14: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.3

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Simplified 97.4%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 6.6

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

11. Simplified 6.6%

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

12. Final simplification 6.6%

    \[\leadsto x \cdot \left(\varepsilon \cdot x\right) \]
13. Add Preprocessing

Alternative 15: 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 63.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 97.4

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

8. Simplified 97.4%

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

9. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. distribute-rgt-out N/A

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

    5. lower-*.f64 N/A

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

    6. lower-fma.f64 97.2

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

11. Simplified 97.2%

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

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot x} \]
13. 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

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

14. Simplified 5.8%

    \[\leadsto \color{blue}{x \cdot \left(\varepsilon \cdot \varepsilon\right)} \]
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: 62.8% 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: 98.9% 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 2024215 
(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)))