2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.5%

Time: 8.9s

Alternatives: 15

Speedup: 76.4×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 62.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(-x\right)\\ t_1 := {\tan x}^{2}\\ t_2 := t\_1 + 1\\ t_3 := t\_2 \cdot \tan x\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_2, -0.5, 0.16666666666666666 \cdot t\_1\right) - t\_2 \cdot t\_1\right) - -0.16666666666666666, \tan x, t\_3 \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left(t\_2, t\_1, -0.16666666666666666 \cdot t\_1 - t\_2 \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, t\_3\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{1 - t\_0 \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (- x)))
        (t_1 (pow (tan x) 2.0))
        (t_2 (+ t_1 1.0))
        (t_3 (* t_2 (tan x))))
   (fma
    (fma
     (fma
      (fma
       (-
        (fma
         (-
          (- (fma t_2 -0.5 (* 0.16666666666666666 t_1)) (* t_2 t_1))
          -0.16666666666666666)
         (tan x)
         (* t_3 -0.3333333333333333)))
       eps
       (-
        (fma t_2 t_1 (- (* -0.16666666666666666 t_1) (* t_2 -0.5)))
        0.16666666666666666))
      eps
      t_3)
     eps
     1.0)
    eps
    (/ (* eps (pow (sin x) 2.0)) (- 1.0 (* t_0 t_0))))))

C

double code(double x, double eps) {
	double t_0 = sin(-x);
	double t_1 = pow(tan(x), 2.0);
	double t_2 = t_1 + 1.0;
	double t_3 = t_2 * tan(x);
	return fma(fma(fma(fma(-fma(((fma(t_2, -0.5, (0.16666666666666666 * t_1)) - (t_2 * t_1)) - -0.16666666666666666), tan(x), (t_3 * -0.3333333333333333)), eps, (fma(t_2, t_1, ((-0.16666666666666666 * t_1) - (t_2 * -0.5))) - 0.16666666666666666)), eps, t_3), eps, 1.0), eps, ((eps * pow(sin(x), 2.0)) / (1.0 - (t_0 * t_0))));
}

Julia

function code(x, eps)
	t_0 = sin(Float64(-x))
	t_1 = tan(x) ^ 2.0
	t_2 = Float64(t_1 + 1.0)
	t_3 = Float64(t_2 * tan(x))
	return fma(fma(fma(fma(Float64(-fma(Float64(Float64(fma(t_2, -0.5, Float64(0.16666666666666666 * t_1)) - Float64(t_2 * t_1)) - -0.16666666666666666), tan(x), Float64(t_3 * -0.3333333333333333))), eps, Float64(fma(t_2, t_1, Float64(Float64(-0.16666666666666666 * t_1) - Float64(t_2 * -0.5))) - 0.16666666666666666)), eps, t_3), eps, 1.0), eps, Float64(Float64(eps * (sin(x) ^ 2.0)) / Float64(1.0 - Float64(t_0 * t_0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[((-N[(N[(N[(N[(t$95$2 * -0.5 + N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] - -0.16666666666666666), $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(t$95$3 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]) * eps + N[(N[(t$95$2 * t$95$1 + N[(N[(-0.16666666666666666 * t$95$1), $MachinePrecision] - N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-cos.f64 99.5

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]
10. Step-by-step derivation

    1. lift-cos.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow2 N/A

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

    4. cos-neg-rev N/A

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

    5. cos-neg-rev N/A

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

    6. 1-sub-sin-rev N/A

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

    7. lower--.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower-sin.f64 N/A

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

    10. lower-neg.f64 N/A

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

    11. lower-sin.f64 N/A

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

    12. lower-neg.f64 99.5

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{1 - \sin \left(-x\right) \cdot \sin \left(-x\right)}\right) \]

11. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{1 - \sin \left(-x\right) \cdot \sin \left(-x\right)}\right) \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 - -1\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_1, t\_0, \mathsf{fma}\left(t\_0, -0.16666666666666666, \left(-1 - t\_0\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_1, 0.16666666666666666 \cdot t\_0\right) - \mathsf{fma}\left(t\_1, t\_0, -0.16666666666666666\right), \tan x, t\_1 \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, t\_1 \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (- t_0 -1.0)))
   (fma
    (fma
     (fma
      (-
       (-
        (fma t_1 t_0 (fma t_0 -0.16666666666666666 (* (- -1.0 t_0) -0.5)))
        0.16666666666666666)
       (*
        (fma
         (-
          (fma -0.5 t_1 (* 0.16666666666666666 t_0))
          (fma t_1 t_0 -0.16666666666666666))
         (tan x)
         (* t_1 (* (tan x) -0.3333333333333333)))
        eps))
      eps
      (* t_1 (tan x)))
     eps
     1.0)
    eps
    (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 - -1.0;
	return fma(fma(fma(((fma(t_1, t_0, fma(t_0, -0.16666666666666666, ((-1.0 - t_0) * -0.5))) - 0.16666666666666666) - (fma((fma(-0.5, t_1, (0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), (t_1 * (tan(x) * -0.3333333333333333))) * eps)), eps, (t_1 * tan(x))), eps, 1.0), eps, ((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 - -1.0)
	return fma(fma(fma(Float64(Float64(fma(t_1, t_0, fma(t_0, -0.16666666666666666, Float64(Float64(-1.0 - t_0) * -0.5))) - 0.16666666666666666) - Float64(fma(Float64(fma(-0.5, t_1, Float64(0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), Float64(t_1 * Float64(tan(x) * -0.3333333333333333))) * eps)), eps, Float64(t_1 * tan(x))), eps, 1.0), eps, Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(t$95$1 * t$95$0 + N[(t$95$0 * -0.16666666666666666 + N[(N[(-1.0 - t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - N[(N[(N[(N[(-0.5 * t$95$1 + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(t$95$1 * N[(N[Tan[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-cos.f64 99.5

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]

11. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.16666666666666666, \left(-1 - {\tan x}^{2}\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, {\tan x}^{2} - -1, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, -0.16666666666666666\right), \tan x, \left({\tan x}^{2} - -1\right) \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, \left({\tan x}^{2} - -1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]
12. Add Preprocessing

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 + 1\\ t_2 := t\_1 \cdot \tan x\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_1, -0.5, 0.16666666666666666 \cdot t\_0\right) - t\_1 \cdot t\_0\right) - -0.16666666666666666, \tan x, t\_2 \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left(t\_1, t\_0, -0.16666666666666666 \cdot t\_0 - t\_1 \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, t\_2\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (+ t_0 1.0)) (t_2 (* t_1 (tan x))))
   (fma
    (fma
     (fma
      (fma
       (-
        (fma
         (-
          (- (fma t_1 -0.5 (* 0.16666666666666666 t_0)) (* t_1 t_0))
          -0.16666666666666666)
         (tan x)
         (* t_2 -0.3333333333333333)))
       eps
       (-
        (fma t_1 t_0 (- (* -0.16666666666666666 t_0) (* t_1 -0.5)))
        0.16666666666666666))
      eps
      t_2)
     eps
     1.0)
    eps
    (/ (* eps (pow (sin x) 2.0)) (+ 0.5 (* 0.5 (cos (* 2.0 x))))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = t_1 * tan(x);
	return fma(fma(fma(fma(-fma(((fma(t_1, -0.5, (0.16666666666666666 * t_0)) - (t_1 * t_0)) - -0.16666666666666666), tan(x), (t_2 * -0.3333333333333333)), eps, (fma(t_1, t_0, ((-0.16666666666666666 * t_0) - (t_1 * -0.5))) - 0.16666666666666666)), eps, t_2), eps, 1.0), eps, ((eps * pow(sin(x), 2.0)) / (0.5 + (0.5 * cos((2.0 * x))))));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(t_1 * tan(x))
	return fma(fma(fma(fma(Float64(-fma(Float64(Float64(fma(t_1, -0.5, Float64(0.16666666666666666 * t_0)) - Float64(t_1 * t_0)) - -0.16666666666666666), tan(x), Float64(t_2 * -0.3333333333333333))), eps, Float64(fma(t_1, t_0, Float64(Float64(-0.16666666666666666 * t_0) - Float64(t_1 * -0.5))) - 0.16666666666666666)), eps, t_2), eps, 1.0), eps, Float64(Float64(eps * (sin(x) ^ 2.0)) / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * x))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[((-N[(N[(N[(N[(t$95$1 * -0.5 + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] - -0.16666666666666666), $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(t$95$2 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]) * eps + N[(N[(t$95$1 * t$95$0 + N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-cos.f64 99.5

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]

9. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) \]
10. Step-by-step derivation

    1. lift-cos.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow2 N/A

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

    4. sqr-cos-a N/A

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

    5. lower-+.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-cos.f64 N/A

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

    8. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \]

11. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, \frac{\varepsilon \cdot {\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}\right) \]
12. Add Preprocessing

Alternative 4: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 - -1\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_1, t\_0, \mathsf{fma}\left(t\_0, -0.16666666666666666, \left(-1 - t\_0\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.5, 0.16666666666666666 \cdot t\_0\right) - \mathsf{fma}\left(t\_1, t\_0, -0.16666666666666666\right), \tan x, t\_1 \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, t\_1 \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, t\_0 \cdot \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (- t_0 -1.0)))
   (fma
    (fma
     (fma
      (-
       (-
        (fma t_1 t_0 (fma t_0 -0.16666666666666666 (* (- -1.0 t_0) -0.5)))
        0.16666666666666666)
       (*
        (fma
         (-
          (fma t_1 -0.5 (* 0.16666666666666666 t_0))
          (fma t_1 t_0 -0.16666666666666666))
         (tan x)
         (* t_1 (* (tan x) -0.3333333333333333)))
        eps))
      eps
      (* t_1 (tan x)))
     eps
     1.0)
    eps
    (* t_0 eps))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 - -1.0;
	return fma(fma(fma(((fma(t_1, t_0, fma(t_0, -0.16666666666666666, ((-1.0 - t_0) * -0.5))) - 0.16666666666666666) - (fma((fma(t_1, -0.5, (0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), (t_1 * (tan(x) * -0.3333333333333333))) * eps)), eps, (t_1 * tan(x))), eps, 1.0), eps, (t_0 * eps));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 - -1.0)
	return fma(fma(fma(Float64(Float64(fma(t_1, t_0, fma(t_0, -0.16666666666666666, Float64(Float64(-1.0 - t_0) * -0.5))) - 0.16666666666666666) - Float64(fma(Float64(fma(t_1, -0.5, Float64(0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), Float64(t_1 * Float64(tan(x) * -0.3333333333333333))) * eps)), eps, Float64(t_1 * tan(x))), eps, 1.0), eps, Float64(t_0 * eps))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(t$95$1 * t$95$0 + N[(t$95$0 * -0.16666666666666666 + N[(N[(-1.0 - t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - N[(N[(N[(N[(t$95$1 * -0.5 + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(t$95$1 * N[(N[Tan[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + N[(t$95$0 * eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

8. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.16666666666666666, \left(-1 - {\tan x}^{2}\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2} - -1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, -0.16666666666666666\right), \tan x, \left({\tan x}^{2} - -1\right) \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, \left({\tan x}^{2} - -1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, {\tan x}^{2} \cdot \varepsilon\right) \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 - -1\\ \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_1, t\_0, \mathsf{fma}\left(t\_0, -0.16666666666666666, \left(-1 - t\_0\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.5, 0.16666666666666666 \cdot t\_0\right) - \mathsf{fma}\left(t\_1, t\_0, -0.16666666666666666\right), \tan x, t\_1 \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, t\_1 \cdot \tan x\right), \varepsilon, 1\right) + t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (- t_0 -1.0)))
   (*
    eps
    (+
     (fma
      (fma
       (-
        (-
         (fma t_1 t_0 (fma t_0 -0.16666666666666666 (* (- -1.0 t_0) -0.5)))
         0.16666666666666666)
        (*
         (fma
          (-
           (fma t_1 -0.5 (* 0.16666666666666666 t_0))
           (fma t_1 t_0 -0.16666666666666666))
          (tan x)
          (* t_1 (* (tan x) -0.3333333333333333)))
         eps))
       eps
       (* t_1 (tan x)))
      eps
      1.0)
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 - -1.0;
	return eps * (fma(fma(((fma(t_1, t_0, fma(t_0, -0.16666666666666666, ((-1.0 - t_0) * -0.5))) - 0.16666666666666666) - (fma((fma(t_1, -0.5, (0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), (t_1 * (tan(x) * -0.3333333333333333))) * eps)), eps, (t_1 * tan(x))), eps, 1.0) + t_0);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 - -1.0)
	return Float64(eps * Float64(fma(fma(Float64(Float64(fma(t_1, t_0, fma(t_0, -0.16666666666666666, Float64(Float64(-1.0 - t_0) * -0.5))) - 0.16666666666666666) - Float64(fma(Float64(fma(t_1, -0.5, Float64(0.16666666666666666 * t_0)) - fma(t_1, t_0, -0.16666666666666666)), tan(x), Float64(t_1 * Float64(tan(x) * -0.3333333333333333))) * eps)), eps, Float64(t_1 * tan(x))), eps, 1.0) + t_0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, N[(eps * N[(N[(N[(N[(N[(N[(t$95$1 * t$95$0 + N[(t$95$0 * -0.16666666666666666 + N[(N[(-1.0 - t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] - N[(N[(N[(N[(t$95$1 * -0.5 + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(t$95$1 * N[(N[Tan[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

8. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.16666666666666666, \left(-1 - {\tan x}^{2}\right) \cdot -0.5\right)\right) - 0.16666666666666666\right) - \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2} - -1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, -0.16666666666666666\right), \tan x, \left({\tan x}^{2} - -1\right) \cdot \left(\tan x \cdot -0.3333333333333333\right)\right) \cdot \varepsilon, \varepsilon, \left({\tan x}^{2} - -1\right) \cdot \tan x\right), \varepsilon, 1\right) + {\tan x}^{2}\right)} \]
9. Add Preprocessing

Alternative 6: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := -1 \cdot t\_2\\ t_4 := 1 - t\_3\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(-1, \varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{t\_0 \cdot t\_4}{t\_1}, \mathsf{fma}\left(-0.5, t\_4, 0.16666666666666666 \cdot t\_2\right)\right)\right), \frac{\sin x \cdot t\_4}{\cos x}\right)\right) - t\_3\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (* -1.0 t_2))
        (t_4 (- 1.0 t_3)))
   (*
    eps
    (-
     (+
      1.0
      (*
       eps
       (fma
        -1.0
        (*
         eps
         (+
          0.16666666666666666
          (fma
           -1.0
           (/ (* t_0 t_4) t_1)
           (fma -0.5 t_4 (* 0.16666666666666666 t_2)))))
        (/ (* (sin x) t_4) (cos x)))))
     t_3))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = -1.0 * t_2;
	double t_4 = 1.0 - t_3;
	return eps * ((1.0 + (eps * fma(-1.0, (eps * (0.16666666666666666 + fma(-1.0, ((t_0 * t_4) / t_1), fma(-0.5, t_4, (0.16666666666666666 * t_2))))), ((sin(x) * t_4) / cos(x))))) - t_3);
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(-1.0 * t_2)
	t_4 = Float64(1.0 - t_3)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * fma(-1.0, Float64(eps * Float64(0.16666666666666666 + fma(-1.0, Float64(Float64(t_0 * t_4) / t_1), fma(-0.5, t_4, Float64(0.16666666666666666 * t_2))))), Float64(Float64(sin(x) * t_4) / cos(x))))) - t_3))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - t$95$3), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(-1.0 * N[(eps * N[(0.16666666666666666 + N[(-1.0 * N[(N[(t$95$0 * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(-0.5 * t$95$4 + N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$4), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Taylor expanded in eps around 0

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

7. Applied rewrites 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \mathsf{fma}\left(-1, \varepsilon \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right), \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Add Preprocessing

Alternative 7: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 + 1\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left(t\_1, t\_0, -0.16666666666666666 \cdot t\_0 - t\_1 \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, t\_1 \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, t\_0 \cdot \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (+ t_0 1.0)))
   (fma
    (fma
     (fma
      (fma
       (* 0.6666666666666666 x)
       eps
       (-
        (fma t_1 t_0 (- (* -0.16666666666666666 t_0) (* t_1 -0.5)))
        0.16666666666666666))
      eps
      (* t_1 (tan x)))
     eps
     1.0)
    eps
    (* t_0 eps))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 + 1.0;
	return fma(fma(fma(fma((0.6666666666666666 * x), eps, (fma(t_1, t_0, ((-0.16666666666666666 * t_0) - (t_1 * -0.5))) - 0.16666666666666666)), eps, (t_1 * tan(x))), eps, 1.0), eps, (t_0 * eps));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 + 1.0)
	return fma(fma(fma(fma(Float64(0.6666666666666666 * x), eps, Float64(fma(t_1, t_0, Float64(Float64(-0.16666666666666666 * t_0) - Float64(t_1 * -0.5))) - 0.16666666666666666)), eps, Float64(t_1 * tan(x))), eps, 1.0), eps, Float64(t_0 * eps))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, N[(N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * eps + N[(N[(t$95$1 * t$95$0 + N[(N[(-0.16666666666666666 * t$95$0), $MachinePrecision] - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + N[(t$95$0 * eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 99.4

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, {\tan x}^{2} \cdot \varepsilon\right) \]

9. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, {\tan x}^{2} \cdot \varepsilon\right) \]
10. Add Preprocessing

Alternative 8: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := t\_0 - -1\\ \varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left(t\_1, t\_0, \mathsf{fma}\left(t\_0, -0.16666666666666666, \left(-1 - t\_0\right) \cdot -0.5\right)\right) - 0.16666666666666666\right), \varepsilon, t\_1 \cdot \tan x\right), \varepsilon, 1\right) + t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (- t_0 -1.0)))
   (*
    eps
    (+
     (fma
      (fma
       (fma
        (* 0.6666666666666666 x)
        eps
        (-
         (fma t_1 t_0 (fma t_0 -0.16666666666666666 (* (- -1.0 t_0) -0.5)))
         0.16666666666666666))
       eps
       (* t_1 (tan x)))
      eps
      1.0)
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = t_0 - -1.0;
	return eps * (fma(fma(fma((0.6666666666666666 * x), eps, (fma(t_1, t_0, fma(t_0, -0.16666666666666666, ((-1.0 - t_0) * -0.5))) - 0.16666666666666666)), eps, (t_1 * tan(x))), eps, 1.0) + t_0);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(t_0 - -1.0)
	return Float64(eps * Float64(fma(fma(fma(Float64(0.6666666666666666 * x), eps, Float64(fma(t_1, t_0, fma(t_0, -0.16666666666666666, Float64(Float64(-1.0 - t_0) * -0.5))) - 0.16666666666666666)), eps, Float64(t_1 * tan(x))), eps, 1.0) + t_0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, N[(eps * N[(N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * eps + N[(N[(t$95$1 * t$95$0 + N[(t$95$0 * -0.16666666666666666 + N[(N[(-1.0 - t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$1 * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 99.4

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, {\tan x}^{2} \cdot \varepsilon\right) \]

9. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \varepsilon, {\tan x}^{2} \cdot \varepsilon\right) \]

11. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \mathsf{fma}\left({\tan x}^{2} - -1, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.16666666666666666, \left(-1 - {\tan x}^{2}\right) \cdot -0.5\right)\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} - -1\right) \cdot \tan x\right), \varepsilon, 1\right) + {\tan x}^{2}\right)} \]
12. Add Preprocessing

Alternative 9: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.3%

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

Alternative 10: 98.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    0.3333333333333333
    eps
    (* x (+ 1.0 (* 0.6666666666666666 (pow eps 2.0)))))
   eps
   1.0)
  eps
  (* (pow (tan x) 2.0) eps)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 98.8

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

9. Applied rewrites 98.8%

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

Alternative 11: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\left(\mathsf{fma}\left({\tan x}^{2} + 1, -0.5, 0.16666666666666666 \cdot {\tan x}^{2}\right) - \left({\tan x}^{2} + 1\right) \cdot {\tan x}^{2}\right) - -0.16666666666666666, \tan x, \left(\left({\tan x}^{2} + 1\right) \cdot \tan x\right) \cdot -0.3333333333333333\right), \varepsilon, \mathsf{fma}\left({\tan x}^{2} + 1, {\tan x}^{2}, -0.16666666666666666 \cdot {\tan x}^{2} - \left({\tan x}^{2} + 1\right) \cdot -0.5\right) - 0.16666666666666666\right), \varepsilon, \left({\tan x}^{2} + 1\right) \cdot \tan x\right), \varepsilon, 1\right), \color{blue}{\varepsilon}, {\tan x}^{2} \cdot \varepsilon\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 98.8

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

9. Applied rewrites 98.8%

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

Alternative 12: 98.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 98.1

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

7. Applied rewrites 98.1%

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

Alternative 13: 98.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (* x (+ eps x)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (eps + x)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + x)));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 98.1

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x\right)\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x\right)}\right) \]
8. Add Preprocessing

Alternative 14: 97.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(-0.5, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}, \frac{\sin x \cdot \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \mathsf{fma}\left(-1, \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.6%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{0.3333333333333333 \cdot {\varepsilon}^{2}}\right) \]
8. Step-by-step derivation

9. Applied rewrites 97.6%

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

Alternative 15: 97.6% accurate, 76.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.0%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \varepsilon \]
6. Step-by-step derivation

7. Applied rewrites 97.6%

   \[\leadsto \varepsilon \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.7× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 62.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 98.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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 2025142 
(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 c (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform c (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform c (+ eps (* eps (tan x) (tan x))))

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