2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.6%

Time: 15.3s

Alternatives: 13

Speedup: 12.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (* (fma (tan x) (tan x) 1.0) (tan x)))
        (t_2
         (+
          (fma t_0 0.3333333333333333 (pow (tan x) 4.0))
          0.3333333333333333)))
   (fma
    (fma
     (fma
      (fma
       eps
       (fma t_1 0.3333333333333333 (fma t_2 (tan x) (pow (tan x) 3.0)))
       (fma (tan x) (tan x) t_2))
      eps
      t_1)
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = fma(tan(x), tan(x), 1.0) * tan(x);
	double t_2 = fma(t_0, 0.3333333333333333, pow(tan(x), 4.0)) + 0.3333333333333333;
	return fma(fma(fma(fma(eps, fma(t_1, 0.3333333333333333, fma(t_2, tan(x), pow(tan(x), 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(fma(tan(x), tan(x), 1.0) * tan(x))
	t_2 = Float64(fma(t_0, 0.3333333333333333, (tan(x) ^ 4.0)) + 0.3333333333333333)
	return fma(fma(fma(fma(eps, fma(t_1, 0.3333333333333333, fma(t_2, tan(x), (tan(x) ^ 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x, 0.3333333333333333, \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, {\tan x}^{4}\right) + 0.3333333333333333, \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, {\tan x}^{4}\right) + 0.3333333333333333\right)\right), \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \left({\tan x}^{4} + \left(0.3333333333333333 - t\_0 \cdot -0.3333333333333333\right)\right) + t\_0\right), \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (fma
    (fma
     (fma
      (fma
       (* 0.6666666666666666 x)
       eps
       (+
        (+
         (pow (tan x) 4.0)
         (- 0.3333333333333333 (* t_0 -0.3333333333333333)))
        t_0))
      eps
      (/ (* (sin x) (fma (tan x) (tan x) 1.0)) (cos x)))
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return fma(fma(fma(fma((0.6666666666666666 * x), eps, ((pow(tan(x), 4.0) + (0.3333333333333333 - (t_0 * -0.3333333333333333))) + t_0)), eps, ((sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return fma(fma(fma(fma(Float64(0.6666666666666666 * x), eps, Float64(Float64((tan(x) ^ 4.0) + Float64(0.3333333333333333 - Float64(t_0 * -0.3333333333333333))) + t_0)), eps, Float64(Float64(sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * eps + N[(N[(N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision] + N[(0.3333333333333333 - N[(t$95$0 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

13. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \left({\tan x}^{4} + \left(0.3333333333333333 - {\tan x}^{2} \cdot -0.3333333333333333\right)\right) + {\tan x}^{2}\right), \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
14. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.8%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.7%

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

Alternative 6: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.7%

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

Alternative 7: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 8: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\mathsf{fma}\left(0.8333333333333334, x \cdot x, 1\right) \cdot x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    0.3333333333333333
    eps
    (/ (* (fma 0.8333333333333334 (* x x) 1.0) x) (cos x)))
   eps
   (pow (tan x) 2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(0.3333333333333333, eps, ((fma(0.8333333333333334, (x * x), 1.0) * x) / cos(x))), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(0.3333333333333333, eps, Float64(Float64(fma(0.8333333333333334, Float64(x * x), 1.0) * x) / cos(x))), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[(0.8333333333333334 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, \frac{x \cdot \left(1 + \frac{5}{6} \cdot {x}^{2}\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\mathsf{fma}\left(0.8333333333333334, x \cdot x, 1\right) \cdot x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
14. Add Preprocessing

Alternative 9: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    (fma 1.3333333333333333 (* eps eps) 1.0)
    x
    (fma 0.6666666666666666 (pow eps 3.0) eps))
   x
   (* (* eps eps) 0.3333333333333333))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma(1.3333333333333333, (eps * eps), 1.0), x, fma(0.6666666666666666, pow(eps, 3.0), eps)), x, ((eps * eps) * 0.3333333333333333)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(1.3333333333333333, Float64(eps * eps), 1.0), x, fma(0.6666666666666666, (eps ^ 3.0), eps)), x, Float64(Float64(eps * eps) * 0.3333333333333333)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(0.6666666666666666 * N[Power[eps, 3.0], $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 10: 98.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (fma (* eps eps) -0.5 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.6

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

   \[\leadsto \frac{\sin \varepsilon}{1 + \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \varepsilon}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{-0.5}, 1\right)} \]
9. Add Preprocessing

Alternative 11: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.6

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\tan \varepsilon} \]
8. Add Preprocessing

Alternative 12: 98.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.6%

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

Alternative 13: 5.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 62.4

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 62.4

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}} \]
6. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{\sin x}{\cos x} \]

   3. mul0-lft 5.4

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

7. Applied rewrites 5.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(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 (+ eps (* eps (tan x) (tan x))))

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