2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 8.1s

Alternatives: 16

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: 62.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := -t\_0\\ t_2 := 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_3 := 1 - t\_1\\ t_4 := \left(\mathsf{fma}\left(t\_3, -0.5, t\_0 \cdot 0.16666666666666666\right) + \left(-\frac{t\_3 \cdot \left(0.5 - t\_2\right)}{0.5 + t\_2}\right)\right) + 0.16666666666666666\\ t_5 := t\_3 \cdot \sin x\\ t_6 := \frac{t\_5}{\cos x}\\ \left(\mathsf{fma}\left(\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(t\_6, -0.5, \frac{\mathsf{fma}\left(t\_4, \sin x, 0.16666666666666666 \cdot t\_5\right)}{\cos x}\right) - t\_4\right) \cdot \varepsilon - \left(-t\_6\right), \varepsilon, 1\right) - t\_1\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (- t_0))
        (t_2 (* 0.5 (cos (* 2.0 x))))
        (t_3 (- 1.0 t_1))
        (t_4
         (+
          (+
           (fma t_3 -0.5 (* t_0 0.16666666666666666))
           (- (/ (* t_3 (- 0.5 t_2)) (+ 0.5 t_2))))
          0.16666666666666666))
        (t_5 (* t_3 (sin x)))
        (t_6 (/ t_5 (cos x))))
   (*
    (-
     (fma
      (-
       (*
        (-
         (*
          (- eps)
          (fma
           t_6
           -0.5
           (/ (fma t_4 (sin x) (* 0.16666666666666666 t_5)) (cos x))))
         t_4)
        eps)
       (- t_6))
      eps
      1.0)
     t_1)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = -t_0;
	double t_2 = 0.5 * cos((2.0 * x));
	double t_3 = 1.0 - t_1;
	double t_4 = (fma(t_3, -0.5, (t_0 * 0.16666666666666666)) + -((t_3 * (0.5 - t_2)) / (0.5 + t_2))) + 0.16666666666666666;
	double t_5 = t_3 * sin(x);
	double t_6 = t_5 / cos(x);
	return (fma(((((-eps * fma(t_6, -0.5, (fma(t_4, sin(x), (0.16666666666666666 * t_5)) / cos(x)))) - t_4) * eps) - -t_6), eps, 1.0) - t_1) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(-t_0)
	t_2 = Float64(0.5 * cos(Float64(2.0 * x)))
	t_3 = Float64(1.0 - t_1)
	t_4 = Float64(Float64(fma(t_3, -0.5, Float64(t_0 * 0.16666666666666666)) + Float64(-Float64(Float64(t_3 * Float64(0.5 - t_2)) / Float64(0.5 + t_2)))) + 0.16666666666666666)
	t_5 = Float64(t_3 * sin(x))
	t_6 = Float64(t_5 / cos(x))
	return Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(-eps) * fma(t_6, -0.5, Float64(fma(t_4, sin(x), Float64(0.16666666666666666 * t_5)) / cos(x)))) - t_4) * eps) - Float64(-t_6)), eps, 1.0) - t_1) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$3 * -0.5 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-N[(N[(t$95$3 * N[(0.5 - t$95$2), $MachinePrecision]), $MachinePrecision] / N[(0.5 + t$95$2), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[((-eps) * N[(t$95$6 * -0.5 + N[(N[(t$95$4 * N[Sin[x], $MachinePrecision] + N[(0.16666666666666666 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision] * eps), $MachinePrecision] - (-t$95$6)), $MachinePrecision] * eps + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision] * eps), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \cos \left(x + x\right)\\ t_2 := 1 - \left(-t\_0\right)\\ \mathsf{fma}\left(\left(\left(\left(\varepsilon \cdot x\right) \cdot 0.6666666666666666 - \mathsf{fma}\left(-0.5, t\_2, \mathsf{fma}\left(0.16666666666666666, t\_0, -\frac{\left(0.5 - t\_1 \cdot 0.5\right) \cdot t\_2}{\mathsf{fma}\left(t\_1, 0.5, 0.5\right)}\right)\right)\right) - 0.16666666666666666\right) \cdot \varepsilon - \left(-t\_2 \cdot \tan x\right), \varepsilon, t\_2\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (cos (+ x x))) (t_2 (- 1.0 (- t_0))))
   (*
    (fma
     (-
      (*
       (-
        (-
         (* (* eps x) 0.6666666666666666)
         (fma
          -0.5
          t_2
          (fma
           0.16666666666666666
           t_0
           (- (/ (* (- 0.5 (* t_1 0.5)) t_2) (fma t_1 0.5 0.5))))))
        0.16666666666666666)
       eps)
      (- (* t_2 (tan x))))
     eps
     t_2)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = cos((x + x));
	double t_2 = 1.0 - -t_0;
	return fma(((((((eps * x) * 0.6666666666666666) - fma(-0.5, t_2, fma(0.16666666666666666, t_0, -(((0.5 - (t_1 * 0.5)) * t_2) / fma(t_1, 0.5, 0.5))))) - 0.16666666666666666) * eps) - -(t_2 * tan(x))), eps, t_2) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = cos(Float64(x + x))
	t_2 = Float64(1.0 - Float64(-t_0))
	return Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(eps * x) * 0.6666666666666666) - fma(-0.5, t_2, fma(0.16666666666666666, t_0, Float64(-Float64(Float64(Float64(0.5 - Float64(t_1 * 0.5)) * t_2) / fma(t_1, 0.5, 0.5)))))) - 0.16666666666666666) * eps) - Float64(-Float64(t_2 * tan(x)))), eps, t_2) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - (-t$95$0)), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(eps * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] - N[(-0.5 * t$95$2 + N[(0.16666666666666666 * t$95$0 + (-N[(N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$1 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * eps), $MachinePrecision] - (-N[(t$95$2 * N[Tan[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(-{\tan x}^{2}\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, x, 0.6666666666666666 \cdot \varepsilon\right), x, 0.3333333333333333\right) \cdot \varepsilon - \left(-t\_0 \cdot \tan x\right), \varepsilon, t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (- (pow (tan x) 2.0)))))
   (*
    (fma
     (-
      (*
       (fma
        (fma 1.3333333333333333 x (* 0.6666666666666666 eps))
        x
        0.3333333333333333)
       eps)
      (- (* t_0 (tan x))))
     eps
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = 1.0 - -pow(tan(x), 2.0);
	return fma(((fma(fma(1.3333333333333333, x, (0.6666666666666666 * eps)), x, 0.3333333333333333) * eps) - -(t_0 * tan(x))), eps, t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(-(tan(x) ^ 2.0)))
	return Float64(fma(Float64(Float64(fma(fma(1.3333333333333333, x, Float64(0.6666666666666666 * eps)), x, 0.3333333333333333) * eps) - Float64(-Float64(t_0 * tan(x)))), eps, t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(1.3333333333333333 * x + N[(0.6666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * eps), $MachinePrecision] - (-N[(t$95$0 * N[Tan[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

12. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-pow.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     (+
      (fma
       (fma
        (fma 1.8888888888888888 (* eps eps) 1.3333333333333333)
        x
        (* 1.3333333333333333 eps))
       x
       (* 0.6666666666666666 (* eps eps)))
      1.0)
     x
     (* 0.3333333333333333 eps))
    eps
    1.0)
   (- (pow (tan x) 2.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(fma((fma(fma(fma(1.8888888888888888, (eps * eps), 1.3333333333333333), x, (1.3333333333333333 * eps)), x, (0.6666666666666666 * (eps * eps))) + 1.0), x, (0.3333333333333333 * eps)), eps, 1.0) - -pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(fma(fma(fma(1.8888888888888888, Float64(eps * eps), 1.3333333333333333), x, Float64(1.3333333333333333 * eps)), x, Float64(0.6666666666666666 * Float64(eps * eps))) + 1.0), x, Float64(0.3333333333333333 * eps)), eps, 1.0) - Float64(-(tan(x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(N[(1.8888888888888888 * N[(eps * eps), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * x + N[(1.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.9%

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

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.0%

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

Alternative 7: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. lower-tan.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-PI.f64 98.9

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

6. Applied rewrites 98.9%

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

Alternative 8: 98.9% accurate, 1.0× speedup

Math

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

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 - Float64(-(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 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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 98.9

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

4. Applied rewrites 98.9%

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

Alternative 9: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.1333333333333333, x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x, 1\right) - \left(-{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x\right)}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    eps
    (* (fma (fma 1.1333333333333333 (* x x) 1.3333333333333333) (* x x) 1.0) x)
    1.0)
   (-
    (pow
     (*
      (fma
       (fma
        (fma 0.05396825396825397 (* x x) 0.13333333333333333)
        (* x x)
        0.3333333333333333)
       (* x x)
       1.0)
      x)
     2.0)))
  eps))

C

double code(double x, double eps) {
	return (fma(eps, (fma(fma(1.1333333333333333, (x * x), 1.3333333333333333), (x * x), 1.0) * x), 1.0) - -pow((fma(fma(fma(0.05396825396825397, (x * x), 0.13333333333333333), (x * x), 0.3333333333333333), (x * x), 1.0) * x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(eps, Float64(fma(fma(1.1333333333333333, Float64(x * x), 1.3333333333333333), Float64(x * x), 1.0) * x), 1.0) - Float64(-(Float64(fma(fma(fma(0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), 0.3333333333333333), Float64(x * x), 1.0) * x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * N[(N[(N[(1.1333333333333333 * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] - (-N[Power[N[(N[(N[(N[(0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 98.5

       \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.1333333333333333, x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x, 1\right) - \left(-{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x\right)}^{2}\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.1333333333333333, x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x, 1\right) - \left(-{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 1\right) \cdot x\right)}^{2}\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 10: 98.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.1333333333333333, x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x, 1\right) - \left(\left(\left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) \cdot \left(x \cdot x\right) - 0.6666666666666666\right) \cdot \left(x \cdot x\right) - 1\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    eps
    (* (fma (fma 1.1333333333333333 (* x x) 1.3333333333333333) (* x x) 1.0) x)
    1.0)
   (*
    (-
     (*
      (-
       (* (- (* -0.19682539682539682 (* x x)) 0.37777777777777777) (* x x))
       0.6666666666666666)
      (* x x))
     1.0)
    (* x x)))
  eps))

C

double code(double x, double eps) {
	return (fma(eps, (fma(fma(1.1333333333333333, (x * x), 1.3333333333333333), (x * x), 1.0) * x), 1.0) - (((((((-0.19682539682539682 * (x * x)) - 0.37777777777777777) * (x * x)) - 0.6666666666666666) * (x * x)) - 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(eps, Float64(fma(fma(1.1333333333333333, Float64(x * x), 1.3333333333333333), Float64(x * x), 1.0) * x), 1.0) - Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.19682539682539682 * Float64(x * x)) - 0.37777777777777777) * Float64(x * x)) - 0.6666666666666666) * Float64(x * x)) - 1.0) * Float64(x * x))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * N[(N[(N[(1.1333333333333333 * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(N[(N[(N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.37777777777777777), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{15}, x \cdot x, \frac{4}{3}\right), x \cdot x, 1\right) \cdot x, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{15}, x \cdot x, \frac{4}{3}\right), x \cdot x, 1\right) \cdot x, 1\right) - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{15}, x \cdot x, \frac{4}{3}\right), x \cdot x, 1\right) \cdot x, 1\right) - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(1.1333333333333333, x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x, 1\right) - \left(\left(\left(-0.19682539682539682 \cdot \left(x \cdot x\right) - 0.37777777777777777\right) \cdot \left(x \cdot x\right) - 0.6666666666666666\right) \cdot \left(x \cdot x\right) - 1\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 11: 98.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 98.4%

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

Alternative 12: 98.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)\right)\right) \cdot x + 1\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon + x \cdot \left(1 + x \cdot \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)\right), x, 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.3%

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

Alternative 13: 98.4% accurate, 11.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 14: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 15: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot {x}^{2} + \varepsilon \]

   2. *-commutative N/A

      \[\leadsto {x}^{2} \cdot \varepsilon + \varepsilon \]

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 98.3

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

7. Applied rewrites 98.3%

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

Alternative 16: 97.9% 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 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-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

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2025101 
(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)))
  (- (tan (+ x eps)) (tan x)))