Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 3.7s

Alternatives: 10

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos \left(x + x\right)\\ t_1 := \frac{0.5 - t\_0}{0.5 + t\_0}\\ \frac{1 - t\_1}{1 + t\_1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (+ x x)))) (t_1 (/ (- 0.5 t_0) (+ 0.5 t_0))))
   (/ (- 1.0 t_1) (+ 1.0 t_1))))

C

double code(double x) {
	double t_0 = 0.5 * cos((x + x));
	double t_1 = (0.5 - t_0) / (0.5 + t_0);
	return (1.0 - t_1) / (1.0 + t_1);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = 0.5d0 * cos((x + x))
    t_1 = (0.5d0 - t_0) / (0.5d0 + t_0)
    code = (1.0d0 - t_1) / (1.0d0 + t_1)
end function

Java

public static double code(double x) {
	double t_0 = 0.5 * Math.cos((x + x));
	double t_1 = (0.5 - t_0) / (0.5 + t_0);
	return (1.0 - t_1) / (1.0 + t_1);
}

Python

def code(x):
	t_0 = 0.5 * math.cos((x + x))
	t_1 = (0.5 - t_0) / (0.5 + t_0)
	return (1.0 - t_1) / (1.0 + t_1)

Julia

function code(x)
	t_0 = Float64(0.5 * cos(Float64(x + x)))
	t_1 = Float64(Float64(0.5 - t_0) / Float64(0.5 + t_0))
	return Float64(Float64(1.0 - t_1) / Float64(1.0 + t_1))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 * cos((x + x));
	t_1 = (0.5 - t_0) / (0.5 + t_0);
	tmp = (1.0 - t_1) / (1.0 + t_1);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 98.9%

   \[\leadsto \frac{1 - \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{{\tan x}^{4}}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (tan x) 4.0)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = sqrt(pow(tan(x), 4.0));
	return (1.0 - t_0) / (1.0 + t_0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((tan(x) ** 4.0d0))
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(Math.pow(Math.tan(x), 4.0));
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.sqrt(math.pow(math.tan(x), 4.0))
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = sqrt((tan(x) ^ 4.0))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((tan(x) ^ 4.0));
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 98.9%

   \[\leadsto \frac{1 - \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. sqr-sin-a-rev N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2 N/A

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

   13. sqr-cos-a-rev N/A

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

   14. frac-times N/A

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

   15. tan-quot N/A

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

   16. lift-tan.f64 N/A

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

   17. tan-quot N/A

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

   18. lift-tan.f64 N/A

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

   19. fabs-sqr N/A

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

   20. lift-*.f64 N/A

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

   21. rem-sqrt-square-rev N/A

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

7. Applied rewrites 98.9%

   \[\leadsto \frac{1 - \color{blue}{\sqrt{{\tan x}^{4}}}}{1 + \frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. sqr-sin-a-rev N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2 N/A

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

   13. sqr-cos-a-rev N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \frac{\sin x \cdot \sin x}{\color{blue}{\cos x \cdot \cos x}}} \]

   14. frac-times N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}} \]

   15. tan-quot N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}} \]

   16. lift-tan.f64 N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\tan x} \cdot \frac{\sin x}{\cos x}} \]

   17. tan-quot N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \tan x \cdot \color{blue}{\tan x}} \]

   18. lift-tan.f64 N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \tan x \cdot \color{blue}{\tan x}} \]

   19. fabs-sqr N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\left|\tan x \cdot \tan x\right|}} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left|\color{blue}{\tan x \cdot \tan x}\right|} \]

   21. rem-sqrt-square-rev N/A

       \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\sqrt{\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}}} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \color{blue}{\sqrt{{\tan x}^{4}}}} \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

   3. lower-pow.f64 99.5%

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]

3. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{\tan x \cdot \tan x}} \]

   2. pow2 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]

   3. lower-pow.f64 99.5%

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]
6. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ t_1 := -1 - {t\_0}^{2}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \left(-t\_0\right) \cdot 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log t\_0 \cdot 2\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))) (t_1 (- -1.0 (pow t_0 2.0))))
   (if (<= t_0 -0.01)
     (/ (expm1 (* (log (- t_0)) 2.0)) t_1)
     (/ (expm1 (* (log t_0) 2.0)) t_1))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double t_1 = -1.0 - pow(t_0, 2.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = expm1((log(-t_0) * 2.0)) / t_1;
	} else {
		tmp = expm1((log(t_0) * 2.0)) / t_1;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.tan(Math.abs(x));
	double t_1 = -1.0 - Math.pow(t_0, 2.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = Math.expm1((Math.log(-t_0) * 2.0)) / t_1;
	} else {
		tmp = Math.expm1((Math.log(t_0) * 2.0)) / t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(math.fabs(x))
	t_1 = -1.0 - math.pow(t_0, 2.0)
	tmp = 0
	if t_0 <= -0.01:
		tmp = math.expm1((math.log(-t_0) * 2.0)) / t_1
	else:
		tmp = math.expm1((math.log(t_0) * 2.0)) / t_1
	return tmp

Julia

function code(x)
	t_0 = tan(abs(x))
	t_1 = Float64(-1.0 - (t_0 ^ 2.0))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(expm1(Float64(log(Float64(-t_0)) * 2.0)) / t_1);
	else
		tmp = Float64(expm1(Float64(log(t_0) * 2.0)) / t_1);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(Exp[N[(N[Log[(-t$95$0)], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(Exp[N[(N[Log[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -0.01

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{-1 - {\tan x}^{2}} \]

      2. lower-unsound-log.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{-1 - {\tan x}^{2}} \]

      3. lower-unsound-expm1.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2} - 1}}{-1 - {\tan x}^{2}} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

      5. pow2 N/A

         \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

      6. sqr-neg-rev N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(\tan x\right)\right)}^{2}} - 1}{-1 - {\tan x}^{2}} \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{neg}\left(\tan x\right)\right) \cdot 2}} - 1}{-1 - {\tan x}^{2}} \]

      9. lower-unsound-expm1.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left(\mathsf{neg}\left(\tan x\right)\right) \cdot 2\right)}}{-1 - {\tan x}^{2}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \left(\mathsf{neg}\left(\tan x\right)\right) \cdot 2}\right)}{-1 - {\tan x}^{2}} \]

      11. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \left(\mathsf{neg}\left(\tan x\right)\right)} \cdot 2\right)}{-1 - {\tan x}^{2}} \]

      12. lower-neg.f64 49.8%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \color{blue}{\left(-\tan x\right)} \cdot 2\right)}{-1 - {\tan x}^{2}} \]

   5. Applied rewrites 49.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \left(-\tan x\right) \cdot 2\right)}}{-1 - {\tan x}^{2}} \]

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, \left|x\right| \cdot \left|x\right|, 1\right) \cdot \left|x\right|\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\frac{-1}{-1 - {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log t\_0 \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\left|x\right| + \left|x\right|\right), -0.5, -0.5\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))))
   (if (<= t_0 -1.0)
     (-
      (tanh
       (log (* (fma 0.3333333333333333 (* (fabs x) (fabs x)) 1.0) (fabs x)))))
     (if (<= t_0 -0.01)
       (/ -1.0 (- -1.0 (pow t_0 2.0)))
       (*
        (expm1 (* (log t_0) 2.0))
        (fma (cos (+ (fabs x) (fabs x))) -0.5 -0.5))))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = -tanh(log((fma(0.3333333333333333, (fabs(x) * fabs(x)), 1.0) * fabs(x))));
	} else if (t_0 <= -0.01) {
		tmp = -1.0 / (-1.0 - pow(t_0, 2.0));
	} else {
		tmp = expm1((log(t_0) * 2.0)) * fma(cos((fabs(x) + fabs(x))), -0.5, -0.5);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(abs(x))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(-tanh(log(Float64(fma(0.3333333333333333, Float64(abs(x) * abs(x)), 1.0) * abs(x)))));
	elseif (t_0 <= -0.01)
		tmp = Float64(-1.0 / Float64(-1.0 - (t_0 ^ 2.0)));
	else
		tmp = Float64(expm1(Float64(log(t_0) * 2.0)) * fma(cos(Float64(abs(x) + abs(x))), -0.5, -0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], (-N[Tanh[N[Log[N[(N[(0.3333333333333333 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, -0.01], N[(-1.0 / N[(-1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[(N[Log[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 x) < -1

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.8%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   6. Applied rewrites 50.8%

      \[\leadsto \left(\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - 1}{-1 - {\tan x}^{2}} \]

   9. Applied rewrites 50.7%

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.8%

          \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2}} \]

   12. Applied rewrites 50.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   14. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   15. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   16. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) - -1\right) \cdot \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   17. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   18. Applied rewrites 50.8%

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

   20. Applied rewrites 27.5%

       \[\leadsto \color{blue}{-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right) \cdot x\right)} \]

   if -1 < (tan.f64 x) < -0.01

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1}}{-1 - {\tan x}^{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{-1}}{-1 - {\tan x}^{2}} \]

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   5. Applied rewrites 98.8%

      \[\leadsto \left(\tan x - -1\right) \cdot \frac{\tan x - 1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(x + x\right), -0.5\right)}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\frac{1 - t\_0 \cdot t\_0}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + 0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log t\_0 \cdot 2\right)}{-1 - {t\_0}^{2}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))))
   (if (<= t_0 -0.01)
     (/ (- 1.0 (* t_0 t_0)) (fma (- 0.5 0.5) (/ 1.0 (+ 0.5 0.5)) 1.0))
     (/ (expm1 (* (log t_0) 2.0)) (- -1.0 (pow t_0 2.0))))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = (1.0 - (t_0 * t_0)) / fma((0.5 - 0.5), (1.0 / (0.5 + 0.5)), 1.0);
	} else {
		tmp = expm1((log(t_0) * 2.0)) / (-1.0 - pow(t_0, 2.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(abs(x))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / fma(Float64(0.5 - 0.5), Float64(1.0 / Float64(0.5 + 0.5)), 1.0));
	else
		tmp = Float64(expm1(Float64(log(t_0) * 2.0)) / Float64(-1.0 - (t_0 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - 0.5), $MachinePrecision] * N[(1.0 / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[(N[Log[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / N[(-1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -0.01

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. frac-times N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - \color{blue}{\frac{1}{2}}, \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, 1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 59.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + \color{blue}{\frac{1}{2}}}, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + \color{blue}{0.5}}, 1\right)} \]

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\frac{1 - t\_0 \cdot t\_0}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + 0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log t\_0 \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\left|x\right| + \left|x\right|\right), -0.5, -0.5\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))))
   (if (<= t_0 -0.01)
     (/ (- 1.0 (* t_0 t_0)) (fma (- 0.5 0.5) (/ 1.0 (+ 0.5 0.5)) 1.0))
     (*
      (expm1 (* (log t_0) 2.0))
      (fma (cos (+ (fabs x) (fabs x))) -0.5 -0.5)))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = (1.0 - (t_0 * t_0)) / fma((0.5 - 0.5), (1.0 / (0.5 + 0.5)), 1.0);
	} else {
		tmp = expm1((log(t_0) * 2.0)) * fma(cos((fabs(x) + fabs(x))), -0.5, -0.5);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(abs(x))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_0)) / fma(Float64(0.5 - 0.5), Float64(1.0 / Float64(0.5 + 0.5)), 1.0));
	else
		tmp = Float64(expm1(Float64(log(t_0) * 2.0)) * fma(cos(Float64(abs(x) + abs(x))), -0.5, -0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - 0.5), $MachinePrecision] * N[(1.0 / N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[(N[Log[t$95$0], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -0.01

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. frac-times N/A

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

      9. mult-flip N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - \color{blue}{\frac{1}{2}}, \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, 1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 59.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + \color{blue}{\frac{1}{2}}}, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - 0.5, \frac{1}{0.5 + \color{blue}{0.5}}, 1\right)} \]

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   5. Applied rewrites 98.8%

      \[\leadsto \left(\tan x - -1\right) \cdot \frac{\tan x - 1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(x + x\right), -0.5\right)}}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 1:\\ \;\;\;\;\frac{-1}{-1 - {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, \left|x\right| \cdot \left|x\right|, 1\right) \cdot \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))))
   (if (<= (* t_0 t_0) 1.0)
     (/ -1.0 (- -1.0 (pow t_0 2.0)))
     (-
      (tanh
       (log
        (* (fma 0.3333333333333333 (* (fabs x) (fabs x)) 1.0) (fabs x))))))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double tmp;
	if ((t_0 * t_0) <= 1.0) {
		tmp = -1.0 / (-1.0 - pow(t_0, 2.0));
	} else {
		tmp = -tanh(log((fma(0.3333333333333333, (fabs(x) * fabs(x)), 1.0) * fabs(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(abs(x))
	tmp = 0.0
	if (Float64(t_0 * t_0) <= 1.0)
		tmp = Float64(-1.0 / Float64(-1.0 - (t_0 ^ 2.0)));
	else
		tmp = Float64(-tanh(log(Float64(fma(0.3333333333333333, Float64(abs(x) * abs(x)), 1.0) * abs(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * t$95$0), $MachinePrecision], 1.0], N[(-1.0 / N[(-1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Tanh[N[Log[N[(N[(0.3333333333333333 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1}}{-1 - {\tan x}^{2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{-1}}{-1 - {\tan x}^{2}} \]

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.8%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   6. Applied rewrites 50.8%

      \[\leadsto \left(\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - 1}{-1 - {\tan x}^{2}} \]

   9. Applied rewrites 50.7%

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.8%

          \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2}} \]

   12. Applied rewrites 50.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   14. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   15. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   16. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) - -1\right) \cdot \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   17. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   18. Applied rewrites 50.8%

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

   20. Applied rewrites 27.5%

       \[\leadsto \color{blue}{-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left|x\right|\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, \left|x\right| \cdot \left|x\right|, 1\right) \cdot \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (tan (fabs x))))
   (if (<= (* t_0 t_0) 1.0)
     1.0
     (-
      (tanh
       (log
        (* (fma 0.3333333333333333 (* (fabs x) (fabs x)) 1.0) (fabs x))))))))

C

double code(double x) {
	double t_0 = tan(fabs(x));
	double tmp;
	if ((t_0 * t_0) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = -tanh(log((fma(0.3333333333333333, (fabs(x) * fabs(x)), 1.0) * fabs(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = tan(abs(x))
	tmp = 0.0
	if (Float64(t_0 * t_0) <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(-tanh(log(Float64(fma(0.3333333333333333, Float64(abs(x) * abs(x)), 1.0) * abs(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * t$95$0), $MachinePrecision], 1.0], 1.0, (-N[Tanh[N[Log[N[(N[(0.3333333333333333 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f32 N/A

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

      7. lower-unsound--.f32 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

          \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      11. lower-unsound-expm1.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      12. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      13. lower-unsound-log.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

      15. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

      17. sub-flip-reverse N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      18. lower--.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

      20. pow2 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

      21. lower-pow.f64 49.5%

          \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
   4. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2} - 1}}{-1 - {\tan x}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \tan x \cdot 2}} - 1}{-1 - {\tan x}^{2}} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{e^{\color{blue}{\log \tan x} \cdot 2} - 1}{-1 - {\tan x}^{2}} \]

      4. exp-to-pow N/A

         \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

      5. pow2 N/A

         \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

      7. lower--.f64 99.5%

         \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x - 1}}{-1 - {\tan x}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

      9. pow2 N/A

         \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

      10. lift-pow.f64 99.5%

          \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{\color{blue}{{\tan x}^{2} - 1}}{-1 - {\tan x}^{2}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto \color{blue}{1} \]

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-sqr-1 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.8%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   6. Applied rewrites 50.8%

      \[\leadsto \left(\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - -1\right) \cdot \frac{\tan x - 1}{-1 - {\tan x}^{2}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7%

         \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right) - 1}{-1 - {\tan x}^{2}} \]

   9. Applied rewrites 50.7%

      \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{\color{blue}{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)} - 1}{-1 - {\tan x}^{2}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 50.8%

          \[\leadsto \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2}} \]

   12. Applied rewrites 50.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   14. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - -1\right) \cdot \frac{x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   15. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   16. Applied rewrites 50.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, 0.3333333333333333, x\right) - -1\right) \cdot \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \color{blue}{0.3333333333333333}, x\right) - 1}{-1 - {\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}^{2}} \]
   17. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

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

       15. pow3 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 50.8%

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

   18. Applied rewrites 50.8%

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

   20. Applied rewrites 27.5%

       \[\leadsto \color{blue}{-\tanh \log \left(\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right) \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.1% accurate, 164.8× speedup

Math

\[1 \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. lower--.f32 N/A

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

   7. lower-unsound--.f32 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. pow-to-exp N/A

       \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2}} - 1}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

   11. lower-unsound-expm1.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

   12. lower-unsound-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

   13. lower-unsound-log.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{\mathsf{neg}\left(\left(1 + \tan x \cdot \tan x\right)\right)} \]

   14. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right)} \]

   15. distribute-neg-in N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1} + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)} \]

   17. sub-flip-reverse N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

   18. lower--.f64 49.5%

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{-1 - \tan x \cdot \tan x}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{\tan x \cdot \tan x}} \]

   20. pow2 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

   21. lower-pow.f64 49.5%

       \[\leadsto \frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

3. Applied rewrites 49.5%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
4. Step-by-step derivation

   1. lift-expm1.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{\log \tan x \cdot 2} - 1}}{-1 - {\tan x}^{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{\log \tan x \cdot 2}} - 1}{-1 - {\tan x}^{2}} \]

   3. lift-log.f64 N/A

      \[\leadsto \frac{e^{\color{blue}{\log \tan x} \cdot 2} - 1}{-1 - {\tan x}^{2}} \]

   4. exp-to-pow N/A

      \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

   5. pow2 N/A

      \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

   7. lower--.f64 99.5%

      \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x - 1}}{-1 - {\tan x}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]

   9. pow2 N/A

      \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

   10. lift-pow.f64 99.5%

       \[\leadsto \frac{\color{blue}{{\tan x}^{2}} - 1}{-1 - {\tan x}^{2}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{{\tan x}^{2} - 1}}{-1 - {\tan x}^{2}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 55.1%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025204 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))