2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.1%

Time: 9.1s

Alternatives: 22

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: 62.4% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \frac{\left(1 + t\_0\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) + t\_0\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    (+
     (fma
      (fma
       (- eps)
       (-
        (* (- (* -1.8888888888888888 (* x x)) 1.3333333333333333) (* x x))
        0.3333333333333333)
       (/ (* (+ 1.0 t_0) (sin x)) (cos x)))
      eps
      1.0)
     t_0)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (fma(fma(-eps, ((((-1.8888888888888888 * (x * x)) - 1.3333333333333333) * (x * x)) - 0.3333333333333333), (((1.0 + t_0) * sin(x)) / cos(x))), eps, 1.0) + t_0) * eps;
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(fma(fma(Float64(-eps), Float64(Float64(Float64(Float64(-1.8888888888888888 * Float64(x * x)) - 1.3333333333333333) * Float64(x * x)) - 0.3333333333333333), Float64(Float64(Float64(1.0 + t_0) * sin(x)) / cos(x))), eps, 1.0) + t_0) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[((-eps) * N[(N[(N[(N[(-1.8888888888888888 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    (+
     (fma
      (fma
       (- eps)
       (- (* (* x x) -1.3333333333333333) 0.3333333333333333)
       (/ (* (+ 1.0 t_0) (sin x)) (cos x)))
      eps
      1.0)
     t_0)
    eps)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (tan(x) ^ 2.0))
	return Float64(fma(fma(t_0, tan(x), Float64(0.3333333333333333 * eps)), 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] + N[(0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in x around 0

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

    1. lower-*.f64 99.9

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

13. Applied rewrites 99.9%

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

14. Final simplification 99.9%

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

Alternative 4: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \frac{\left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     (- eps)
     (-
      (* (- (* -1.8888888888888888 (* x x)) 1.3333333333333333) (* x x))
      0.3333333333333333)
     (/
      (*
       (+
        1.0
        (*
         (fma
          (fma
           (fma 0.19682539682539682 (* x x) 0.37777777777777777)
           (* x x)
           0.6666666666666666)
          (* x x)
          1.0)
         (* x x)))
       (sin x))
      (cos x)))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(-eps, ((((-1.8888888888888888 * (x * x)) - 1.3333333333333333) * (x * x)) - 0.3333333333333333), (((1.0 + (fma(fma(fma(0.19682539682539682, (x * x), 0.37777777777777777), (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * sin(x)) / cos(x))), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(-eps), Float64(Float64(Float64(Float64(-1.8888888888888888 * Float64(x * x)) - 1.3333333333333333) * Float64(x * x)) - 0.3333333333333333), Float64(Float64(Float64(1.0 + Float64(fma(fma(fma(0.19682539682539682, Float64(x * x), 0.37777777777777777), Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x))) * sin(x)) / cos(x))), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, 1 \cdot \frac{\left(1 - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \frac{\left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 5: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2748015873015873, x \cdot x, 0.5083333333333333\right), x \cdot x, 0.8333333333333334\right), x \cdot x, 1\right) \cdot x}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     (- eps)
     (-
      (* (- (* -1.8888888888888888 (* x x)) 1.3333333333333333) (* x x))
      0.3333333333333333)
     (/
      (*
       (fma
        (fma
         (fma 0.2748015873015873 (* x x) 0.5083333333333333)
         (* x x)
         0.8333333333333334)
        (* x x)
        1.0)
       x)
      (cos x)))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(-eps, ((((-1.8888888888888888 * (x * x)) - 1.3333333333333333) * (x * x)) - 0.3333333333333333), ((fma(fma(fma(0.2748015873015873, (x * x), 0.5083333333333333), (x * x), 0.8333333333333334), (x * x), 1.0) * x) / cos(x))), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(-eps), Float64(Float64(Float64(Float64(-1.8888888888888888 * Float64(x * x)) - 1.3333333333333333) * Float64(x * x)) - 0.3333333333333333), Float64(Float64(fma(fma(fma(0.2748015873015873, Float64(x * x), 0.5083333333333333), Float64(x * x), 0.8333333333333334), Float64(x * x), 1.0) * x) / cos(x))), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[((-eps) * N[(N[(N[(N[(-1.8888888888888888 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + N[(N[(N[(N[(N[(0.2748015873015873 * N[(x * x), $MachinePrecision] + 0.5083333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.8333333333333334), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, 1 \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{5}{6} + {x}^{2} \cdot \left(\frac{61}{120} + \frac{277}{1008} \cdot {x}^{2}\right)\right)\right)}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, 1 \cdot \frac{\left(1 + {x}^{2} \cdot \left(\frac{5}{6} + {x}^{2} \cdot \left(\frac{61}{120} + \frac{277}{1008} \cdot {x}^{2}\right)\right)\right) \cdot x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, 1 \cdot \frac{\left(1 + {x}^{2} \cdot \left(\frac{5}{6} + {x}^{2} \cdot \left(\frac{61}{120} + \frac{277}{1008} \cdot {x}^{2}\right)\right)\right) \cdot x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, 1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2748015873015873, x \cdot x, 0.5083333333333333\right), x \cdot x, 0.8333333333333334\right), x \cdot x, 1\right) \cdot x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.2748015873015873, x \cdot x, 0.5083333333333333\right), x \cdot x, 0.8333333333333334\right), x \cdot x, 1\right) \cdot x}{\cos x}\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 6: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.7873015873015873, x \cdot x, 1.1333333333333333\right), x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     (- eps)
     (-
      (* (- (* -1.8888888888888888 (* x x)) 1.3333333333333333) (* x x))
      0.3333333333333333)
     (*
      (fma
       (fma
        (fma 0.7873015873015873 (* x x) 1.1333333333333333)
        (* x x)
        1.3333333333333333)
       (* x x)
       1.0)
      x))
    eps
    1.0)
   (pow (tan x) 2.0))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(-eps, ((((-1.8888888888888888 * (x * x)) - 1.3333333333333333) * (x * x)) - 0.3333333333333333), (fma(fma(fma(0.7873015873015873, (x * x), 1.1333333333333333), (x * x), 1.3333333333333333), (x * x), 1.0) * x)), eps, 1.0) + pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(-eps), Float64(Float64(Float64(Float64(-1.8888888888888888 * Float64(x * x)) - 1.3333333333333333) * Float64(x * x)) - 0.3333333333333333), Float64(fma(fma(fma(0.7873015873015873, Float64(x * x), 1.1333333333333333), Float64(x * x), 1.3333333333333333), Float64(x * x), 1.0) * x)), eps, 1.0) + (tan(x) ^ 2.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[((-eps) * N[(N[(N[(N[(-1.8888888888888888 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + N[(N[(N[(N[(0.7873015873015873 * N[(x * x), $MachinePrecision] + 1.1333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, x \cdot \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right)\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right) \cdot x\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(\frac{-17}{9} \cdot \left(x \cdot x\right) - \frac{4}{3}\right) \cdot \left(x \cdot x\right) - \frac{1}{3}, \left(1 + {x}^{2} \cdot \left(\frac{4}{3} + {x}^{2} \cdot \left(\frac{17}{15} + \frac{248}{315} \cdot {x}^{2}\right)\right)\right) \cdot x\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.7873015873015873, x \cdot x, 1.1333333333333333\right), x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

12. Final simplification 99.8%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \left(-1.8888888888888888 \cdot \left(x \cdot x\right) - 1.3333333333333333\right) \cdot \left(x \cdot x\right) - 0.3333333333333333, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.7873015873015873, x \cdot x, 1.1333333333333333\right), x \cdot x, 1.3333333333333333\right), x \cdot x, 1\right) \cdot x\right), \varepsilon, 1\right) + {\tan x}^{2}\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 7: 98.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 99.8%

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

12. Final simplification 99.8%

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

Alternative 8: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 9: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 10: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 11: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (1.0d0 + (tan(x) ** 2.0d0)) * eps
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. frac-times N/A

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

   8. tan-quot N/A

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

   9. tan-quot N/A

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

   10. lower-neg.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-tan.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Final simplification 99.4%

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

Alternative 12: 98.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma (fma (- eps) -0.3333333333333333 (/ x (cos x))) eps 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 (fma(fma(-eps, -0.3333333333333333, (x / cos(x))), eps, 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(fma(fma(Float64(-eps), -0.3333333333333333, Float64(x / cos(x))), eps, 1.0) + 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[(N[(N[((-eps) * -0.3333333333333333 + N[(x / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] + 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.6%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

14. Applied rewrites 99.4%

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

15. Final simplification 99.4%

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

Alternative 13: 98.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma x eps 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(x, eps, 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(x, eps, 1.0) + (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[(x * eps + 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. pow2 N/A

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

    12. lift-*.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 N/A

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

    15. pow2 N/A

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

    16. lift-*.f64 99.3

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

14. Applied rewrites 99.3%

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

15. Final simplification 99.3%

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

Alternative 14: 98.3% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma x eps 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 (fma(x, eps, 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(fma(x, eps, 1.0) + 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[(N[(x * eps + 1.0), $MachinePrecision] + 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

14. Applied rewrites 99.3%

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

15. Final simplification 99.3%

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

Alternative 15: 98.3% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma (fma 0.3333333333333333 eps x) eps 1.0)
   (*
    (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0)
    (* x x)))
  eps))

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.3333333333333333 * eps + x), $MachinePrecision] * eps + 1.0), $MachinePrecision] + N[(N[(N[(0.37777777777777777 * 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 16: 98.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(x * eps + 1.0), $MachinePrecision] - N[(N[(N[(N[(N[(-0.37777777777777777 * 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 99.3

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

14. Applied rewrites 99.3%

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

Alternative 17: 98.2% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   1.0
   (*
    (fma (fma 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(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(1.0 + Float64(fma(fma(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[(0.37777777777777777 * 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 99.1%

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

15. Final simplification 99.1%

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

Alternative 18: 98.2% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

12. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 99.1

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

14. Applied rewrites 99.1%

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

15. Final simplification 99.1%

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

Alternative 19: 98.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 98.9

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

11. Applied rewrites 98.9%

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

Alternative 20: 98.1% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-+.f64 N/A

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

    5. lower-fma.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 98.9

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

11. Applied rewrites 98.9%

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

12. Final simplification 98.9%

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

Alternative 21: 98.0% 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 65.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. pow2 N/A

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

    5. lower-fma.f64 98.8

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

11. Applied rewrites 98.8%

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

12. Taylor expanded in x around 0

    \[\leadsto \varepsilon + \varepsilon \cdot {x}^{\color{blue}{2}} \]
13. 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 98.8

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

14. Applied rewrites 98.8%

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

Alternative 22: 97.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 65.9%

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

3. Taylor expanded in x around 0

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

   1. quot-tan N/A

      \[\leadsto \tan \varepsilon \]

   2. lower-tan.f64 98.2

      \[\leadsto \tan \varepsilon \]

5. Applied rewrites 98.2%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 98.2%

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

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025084 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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