2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.6%

Time: 11.8s

Alternatives: 32

Speedup: 207.0×

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(fma(Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)), -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)
	t_3 = Float64(t_1 * sin(x))
	t_4 = Float64(t_3 / cos(x))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(t_4, -0.5, Float64(fma(t_2, sin(x), Float64(0.16666666666666666 * t_3)) / cos(x)))) - t_2), eps, t_4), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$4 * -0.5 + N[(N[(t$95$2 * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] * eps + t$95$4), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(fma(Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)), -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)
	t_3 = Float64(Float64(t_1 * sin(x)) / cos(x))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(t_3, -0.5, Float64(fma(t_2, sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - t_2), eps, t_3), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$3 * -0.5 + N[(N[(t$95$2 * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0))
	t_3 = Float64(Float64(t_1 * sin(x)) / cos(x))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(t_3, -0.5, Float64(fma(Float64(fma(t_2, -1.0, -0.5) + 0.16666666666666666), sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - Float64(fma(t_2, -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)), eps, t_3), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$3 * -0.5 + N[(N[(N[(N[(t$95$2 * -1.0 + -0.5), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(t_1 * sin(x))
	t_3 = Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(Float64(t_2 / Float64(1.0 + Float64(-0.5 * Float64(x * x)))), -0.5, Float64(fma(Float64(fma(t_3, -1.0, -0.5) + 0.16666666666666666), sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - Float64(fma(t_3, -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)), eps, Float64(t_2 / cos(x))), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(N[(t$95$2 / N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t$95$3 * -1.0 + -0.5), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$3 * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$2 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.8%

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

12. Taylor expanded in x around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 99.8

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

14. Applied rewrites 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0))
	t_3 = Float64(t_1 * sin(x))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(Float64(t_3 / 1.0), -0.5, Float64(fma(Float64(fma(t_2, -1.0, -0.5) + 0.16666666666666666), sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - Float64(fma(t_2, -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)), eps, Float64(t_3 / cos(x))), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(N[(t$95$3 / 1.0), $MachinePrecision] * -0.5 + N[(N[(N[(N[(t$95$2 * -1.0 + -0.5), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$2 * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + N[(t$95$3 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.8%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 99.8%

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

15. Final simplification 99.8%

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

Alternative 6: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(t_1 * sin(x)) / cos(x))
	t_3 = Float64(t_1 * (sin(x) ^ 2.0))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(t_2, -0.5, Float64(fma(Float64(fma(Float64(t_3 / 1.0), -1.0, -0.5) + 0.16666666666666666), sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - Float64(fma(Float64(t_3 / (cos(x) ^ 2.0)), -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)), eps, t_2), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$2 * -0.5 + N[(N[(N[(N[(N[(t$95$3 / 1.0), $MachinePrecision] * -1.0 + -0.5), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$3 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.8%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 99.7%

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

15. Final simplification 99.7%

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

Alternative 7: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(t_1 * sin(x)) / cos(x))
	return Float64(Float64(fma(fma(Float64(Float64(Float64(-eps) * fma(t_2, -0.5, Float64(fma(-0.3333333333333333, sin(x), Float64(0.16666666666666666 * Float64(1.0 * sin(x)))) / cos(x)))) - Float64(fma(Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)), -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666)), eps, t_2), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$2 * -0.5 + N[(N[(-0.3333333333333333 * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * N[(1.0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 8: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   5. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x + \frac{\sin \varepsilon}{\cos \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   7. lift-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x} + \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   8. quot-tan N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   9. lower-tan.f64 N/A

      \[\leadsto \frac{\tan x + \color{blue}{\tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   11. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   13. lift-tan.f64 N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x \]

   14. quot-tan N/A

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

   15. lower-tan.f64 60.0

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} - \tan x \]

4. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

5. Taylor expanded in eps around 0

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 9: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	return Float64(Float64(fma(fma(Float64(-eps), Float64(fma(Float64(Float64(t_1 * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)), -1.0, fma(t_1, -0.5, Float64(t_0 * 0.16666666666666666))) + 0.16666666666666666), Float64(Float64(t_1 * sin(x)) / cos(x))), eps, 1.0) + 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[(1.0 + t$95$0), $MachinePrecision]}, N[(N[(N[(N[((-eps) * N[(N[(N[(N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(t$95$1 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 10: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 99.5

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 11: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 12: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 13: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + t\_0\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) + t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    (+
     (fma
      (fma
       0.3333333333333333
       eps
       (/
        (*
         (+ 1.0 t_0)
         (*
          x
          (+
           1.0
           (*
            (* x x)
            (-
             (*
              (* x x)
              (+ 0.008333333333333333 (* -0.0001984126984126984 (* x x))))
             0.16666666666666666)))))
        (cos x)))
      eps
      1.0)
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (fma(fma(0.3333333333333333, eps, (((1.0 + t_0) * (x * (1.0 + ((x * x) * (((x * x) * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))) - 0.16666666666666666))))) / cos(x))), eps, 1.0) + t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(Float64(Float64(1.0 + t_0) * Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(x * x)))) - 0.16666666666666666))))) / cos(x))), eps, 1.0) + t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    10. lower-+.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    12. unpow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(x \cdot x\right)\right) - \frac{1}{6}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 99.2

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + {\tan x}^{2}\right) \cdot \left(x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right) - 0.16666666666666666\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 14: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + t\_0\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) - 0.5\right)}\right), \varepsilon, 1\right) + t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    (+
     (fma
      (fma
       0.3333333333333333
       eps
       (/
        (* (+ 1.0 t_0) (* x (+ 1.0 (* -0.16666666666666666 (* x x)))))
        (+
         1.0
         (*
          (* x x)
          (-
           (*
            (* x x)
            (+ 0.041666666666666664 (* -0.001388888888888889 (* x x))))
           0.5)))))
      eps
      1.0)
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (fma(fma(0.3333333333333333, eps, (((1.0 + t_0) * (x * (1.0 + (-0.16666666666666666 * (x * x))))) / (1.0 + ((x * x) * (((x * x) * (0.041666666666666664 + (-0.001388888888888889 * (x * x)))) - 0.5))))), eps, 1.0) + t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(Float64(Float64(1.0 + t_0) * Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))))) / Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(0.041666666666666664 + Float64(-0.001388888888888889 * Float64(x * x)))) - 0.5))))), eps, 1.0) + t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 99.1

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

11. Applied rewrites 99.1%

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

12. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    9. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    11. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot \left(x \cdot x\right)\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    12. lift-*.f64 99.2

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) - 0.5\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

14. Applied rewrites 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) - 0.5\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

15. Final simplification 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + {\tan x}^{2}\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(x \cdot x\right)\right) - 0.5\right)}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
16. Add Preprocessing

Alternative 15: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + t\_0\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5\right)}\right), \varepsilon, 1\right) + t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    (+
     (fma
      (fma
       0.3333333333333333
       eps
       (/
        (* (+ 1.0 t_0) (* x (+ 1.0 (* -0.16666666666666666 (* x x)))))
        (+ 1.0 (* (* x x) (- (* 0.041666666666666664 (* x x)) 0.5)))))
      eps
      1.0)
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (fma(fma(0.3333333333333333, eps, (((1.0 + t_0) * (x * (1.0 + (-0.16666666666666666 * (x * x))))) / (1.0 + ((x * x) * ((0.041666666666666664 * (x * x)) - 0.5))))), eps, 1.0) + t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(Float64(Float64(1.0 + t_0) * Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))))) / Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5))))), eps, 1.0) + t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 99.1

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

11. Applied rewrites 99.1%

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

12. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. lift-*.f64 99.2

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

14. Applied rewrites 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5\right)}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

15. Final simplification 99.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 + {\tan x}^{2}\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{1 + \left(x \cdot x\right) \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5\right)}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
16. Add Preprocessing

Alternative 16: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     0.3333333333333333
     eps
     (/
      (*
       (-
        1.0
        (*
         (* x x)
         (-
          (*
           (* x x)
           (-
            (*
             (* x x)
             (- (* -0.19682539682539682 (* x x)) 0.37777777777777777))
            0.6666666666666666))
          1.0)))
       (* x (+ 1.0 (* -0.16666666666666666 (* x x)))))
      (cos x)))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (((1.0 - ((x * x) * (((x * x) * (((x * x) * ((-0.19682539682539682 * (x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * (x * (1.0 + (-0.16666666666666666 * (x * x))))) / cos(x))), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(Float64(Float64(1.0 - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.19682539682539682 * Float64(x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))))) / cos(x))), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.37777777777777777), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 99.1

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

11. Applied rewrites 99.1%

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

12. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    12. lower--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    14. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot \left(x \cdot x\right) - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    15. lift-*.f64 99.0

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

14. Applied rewrites 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, 1 \cdot \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

15. Final simplification 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\left(1 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \left(x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
16. Add Preprocessing

Alternative 17: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + \left(x \cdot x\right) \cdot \left(1.1333333333333333 + 0.7873015873015873 \cdot \left(x \cdot x\right)\right)\right)\right)\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     0.3333333333333333
     eps
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         1.3333333333333333
         (* (* x x) (+ 1.1333333333333333 (* 0.7873015873015873 (* x x)))))))))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + ((x * x) * (1.3333333333333333 + ((x * x) * (1.1333333333333333 + (0.7873015873015873 * (x * x))))))))), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(1.3333333333333333 + Float64(Float64(x * x) * Float64(1.1333333333333333 + Float64(0.7873015873015873 * Float64(x * x))))))))), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.3333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(1.1333333333333333 + N[(0.7873015873015873 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \left(x \cdot x\right) \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \left(x \cdot x\right) \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    10. lower-+.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \left(x \cdot x\right) \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \left(x \cdot x\right) \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    12. unpow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \left(x \cdot x\right) \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot \left(x \cdot x\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 99.0

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + \left(x \cdot x\right) \cdot \left(1.1333333333333333 + 0.7873015873015873 \cdot \left(x \cdot x\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + \left(x \cdot x\right) \cdot \left(1.1333333333333333 + 0.7873015873015873 \cdot \left(x \cdot x\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 99.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + \left(x \cdot x\right) \cdot \left(1.1333333333333333 + 0.7873015873015873 \cdot \left(x \cdot x\right)\right)\right)\right)\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 18: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + 1.1333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     0.3333333333333333
     eps
     (*
      x
      (+
       1.0
       (* (* x x) (+ 1.3333333333333333 (* 1.1333333333333333 (* x x)))))))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + ((x * x) * (1.3333333333333333 + (1.1333333333333333 * (x * x))))))), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(1.3333333333333333 + Float64(1.1333333333333333 * Float64(x * x))))))), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(1.3333333333333333 + N[(1.1333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    4. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    6. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot {x}^{2}\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    8. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(\frac{4}{3} + \frac{17}{15} \cdot \left(x \cdot x\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 98.9

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + 1.1333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + 1.1333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 98.9%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(1.3333333333333333 + 1.1333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 19: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 20: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.9%

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

12. Final simplification 98.9%

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

Alternative 21: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon + x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 98.9

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 98.9%

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

15. Final simplification 98.9%

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

Alternative 22: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (1.0 + pow(tan(x), 2.0)) * 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 = (1.0d0 + (tan(x) ** 2.0d0)) * eps
end function

Java

public static double code(double x, double eps) {
	return (1.0 + Math.pow(Math.tan(x), 2.0)) * eps;
}

Python

def code(x, eps):
	return (1.0 + math.pow(math.tan(x), 2.0)) * eps

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (1.0 + (tan(x) ^ 2.0)) * eps;
end

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. frac-times N/A

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

   8. tan-quot N/A

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

   9. tan-quot N/A

      \[\leadsto \left(1 - \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)\right) \cdot \varepsilon \]

   10. lower-neg.f64 N/A

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

   11. pow2 N/A

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

   13. lift-tan.f64 98.7

       \[\leadsto \left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

   \[\leadsto \left(1 + {\tan x}^{2}\right) \cdot \varepsilon \]
7. Add Preprocessing

Alternative 23: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     0.3333333333333333
     eps
     (*
      x
      (+
       1.0
       (fma
        0.6666666666666666
        (* eps eps)
        (*
         x
         (fma
          1.3333333333333333
          eps
          (* x (+ 1.3333333333333333 (* 1.8888888888888888 (* eps eps))))))))))
    eps
    1.0)
   (*
    (* x x)
    (-
     (*
      (* x x)
      (-
       (* (* x x) (- (* -0.19682539682539682 (* x x)) 0.37777777777777777))
       0.6666666666666666))
     1.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(0.3333333333333333, eps, (x * (1.0 + fma(0.6666666666666666, (eps * eps), (x * fma(1.3333333333333333, eps, (x * (1.3333333333333333 + (1.8888888888888888 * (eps * eps)))))))))), eps, 1.0) - ((x * x) * (((x * x) * (((x * x) * ((-0.19682539682539682 * (x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(0.3333333333333333, eps, Float64(x * Float64(1.0 + fma(0.6666666666666666, Float64(eps * eps), Float64(x * fma(1.3333333333333333, eps, Float64(x * Float64(1.3333333333333333 + Float64(1.8888888888888888 * Float64(eps * eps)))))))))), eps, 1.0) - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.19682539682539682 * Float64(x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + N[(x * N[(1.0 + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision] + N[(x * N[(1.3333333333333333 * eps + N[(x * N[(1.3333333333333333 + N[(1.8888888888888888 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.37777777777777777), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon + x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    2. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    6. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    10. unpow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    11. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    12. lower--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    14. unpow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(\frac{2}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\frac{4}{3}, \varepsilon, x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot \left(x \cdot x\right) - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    15. lower-*.f64 98.2

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 24: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma (* x (+ 1.0 (* 1.3333333333333333 (* x x)))) eps 1.0)
   (*
    (* x x)
    (-
     (*
      (* x x)
      (-
       (* (* x x) (- (* -0.19682539682539682 (* x x)) 0.37777777777777777))
       0.6666666666666666))
     1.0)))
  eps))

C

double code(double x, double eps) {
	return (fma((x * (1.0 + (1.3333333333333333 * (x * x)))), eps, 1.0) - ((x * x) * (((x * x) * (((x * x) * ((-0.19682539682539682 * (x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x * Float64(1.0 + Float64(1.3333333333333333 * Float64(x * x)))), eps, 1.0) - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.19682539682539682 * Float64(x * x)) - 0.37777777777777777)) - 0.6666666666666666)) - 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x * N[(1.0 + N[(1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.37777777777777777), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon + x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 98.9

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    2. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    12. lower--.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    14. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-62}{315} \cdot \left(x \cdot x\right) - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    15. lift-*.f64 98.2

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]

14. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 25: 98.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma (* x (+ 1.0 (* 1.3333333333333333 (* x x)))) eps 1.0)
   (*
    (* x x)
    (-
     (* (* x x) (- (* -0.37777777777777777 (* x x)) 0.6666666666666666))
     1.0)))
  eps))

C

double code(double x, double eps) {
	return (fma((x * (1.0 + (1.3333333333333333 * (x * x)))), eps, 1.0) - ((x * x) * (((x * x) * ((-0.37777777777777777 * (x * x)) - 0.6666666666666666)) - 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x * Float64(1.0 + Float64(1.3333333333333333 * Float64(x * x)))), eps, 1.0) - Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * Float64(Float64(-0.37777777777777777 * Float64(x * x)) - 0.6666666666666666)) - 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x * N[(1.0 + N[(1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(-0.37777777777777777 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon + x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 98.9

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    2. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    8. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot {x}^{2} - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-17}{45} \cdot \left(x \cdot x\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 98.0

        \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]

14. Applied rewrites 98.0%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.37777777777777777 \cdot \left(x \cdot x\right) - 0.6666666666666666\right) - 1\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 26: 98.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} \cdot \varepsilon + x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, \varepsilon, x \cdot \left(1 + \left(\frac{2}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{4}{3} \cdot \varepsilon + x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Applied rewrites 98.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, x \cdot \left(1 + \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(1.3333333333333333, \varepsilon, x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 98.9

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.9%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. pow2 N/A

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

    3. lift-*.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 97.7

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

14. Applied rewrites 97.7%

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), \varepsilon, 1\right) - \left(x \cdot x\right) \cdot \left(-0.6666666666666666 \cdot \left(x \cdot x\right) - 1\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 27: 98.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

   \[\leadsto \left(1 + \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1, x \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 97.5%

    \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1, x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 97.5

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

14. Applied rewrites 97.5%

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

Alternative 28: 98.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f64 97.5

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

11. Applied rewrites 97.5%

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

Alternative 29: 98.2% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 97.5%

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

Alternative 30: 98.2% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

   \[\leadsto \left(1 + \mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1, x \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 97.5%

    \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1, x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

14. Applied rewrites 97.5%

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

Alternative 31: 98.1% accurate, 14.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (1.0 + (x * 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 = (1.0d0 + (x * x)) * eps
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \left(1 + \mathsf{fma}\left(0.3333333333333333, \varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(\varepsilon, 1 + 0.6666666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, \varepsilon \cdot \left(x \cdot \left(1.3333333333333333 + 1.8888888888888888 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 97.4

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

11. Applied rewrites 97.4%

    \[\leadsto \left(1 + x \cdot x\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 32: 97.7% accurate, 207.0× 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 59.9%

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

3. Taylor expanded in x around 0

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

   1. quot-tan N/A

      \[\leadsto \tan \varepsilon \]

   2. lower-tan.f64 96.9

      \[\leadsto \tan \varepsilon \]

5. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\tan \varepsilon} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 96.9%

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

Developer Target 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2025051 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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