Trigonometry B

Percentage Accurate: 99.5% → 99.6%

Time: 4.3s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

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

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

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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := \left(0.5 - t\_0\right) \cdot \frac{1}{0.5 + t\_0}\\ \frac{1 - t\_1}{1 + t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 x))))
        (t_1 (* (- 0.5 t_0) (/ 1.0 (+ 0.5 t_0)))))
   (/ (- 1.0 t_1) (+ 1.0 t_1))))

C

double code(double x) {
	double t_0 = 0.5 * cos((2.0 * x));
	double t_1 = (0.5 - t_0) * (1.0 / (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((2.0d0 * x))
    t_1 = (0.5d0 - t_0) * (1.0d0 / (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((2.0 * x));
	double t_1 = (0.5 - t_0) * (1.0 / (0.5 + t_0));
	return (1.0 - t_1) / (1.0 + t_1);
}

Python

def code(x):
	t_0 = 0.5 * math.cos((2.0 * x))
	t_1 = (0.5 - t_0) * (1.0 / (0.5 + t_0))
	return (1.0 - t_1) / (1.0 + t_1)

Julia

function code(x)
	t_0 = Float64(0.5 * cos(Float64(2.0 * x)))
	t_1 = Float64(Float64(0.5 - t_0) * Float64(1.0 / 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((2.0 * x));
	t_1 = (0.5 - t_0) * (1.0 / (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[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(1.0 / N[(0.5 + t$95$0), $MachinePrecision]), $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. quot-tan 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. quot-tan 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. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.0

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

3. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. quot-tan N/A

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

   4. lift-tan.f64 N/A

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

   5. quot-tan N/A

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

   6. frac-times N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.5

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 - 1.0), $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. quot-tan 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. quot-tan 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. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.0

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

3. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. quot-tan N/A

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

   4. lift-tan.f64 N/A

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

   5. quot-tan N/A

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

   6. frac-times N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.5

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

7. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
8. 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. lift-tan.f64 N/A

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

   5. pow-to-exp N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lower--.f64 99.5

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

9. Applied rewrites 99.5%

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

    1. lift-pow.f64 N/A

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

    2. lift-tan.f64 N/A

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

    3. pow2 N/A

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

    4. quot-tan N/A

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

    5. quot-tan N/A

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

    6. frac-times N/A

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

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

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

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

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

    9. div-sub N/A

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

    10. count-2-rev N/A

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

    11. *-commutative N/A

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

    12. sub-div N/A

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

    13. lift-cos.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. lift--.f64 N/A

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

    17. +-commutative N/A

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

11. Applied rewrites 99.0%

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

    1. lift-pow.f64 N/A

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

    2. lift-tan.f64 N/A

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

    3. pow2 N/A

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

    4. quot-tan N/A

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

    5. quot-tan N/A

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

    6. frac-times N/A

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

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

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

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

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

    9. div-sub N/A

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

    10. count-2-rev N/A

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

    11. *-commutative N/A

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

    12. sub-div N/A

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

    13. lift-cos.f64 N/A

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

    14. lift-+.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. lift--.f64 N/A

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

    17. +-commutative N/A

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

13. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (t_0 - 1.0) / (-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 = (t_0 - 1.0d0) / ((-1.0d0) - t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(t$95$0 - 1.0), $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. quot-tan 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. quot-tan 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. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.0

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

3. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. quot-tan N/A

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

   4. lift-tan.f64 N/A

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

   5. quot-tan N/A

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

   6. frac-times N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.5

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

7. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}} \]
8. 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. lift-tan.f64 N/A

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

   5. pow-to-exp N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lower--.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 4: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\tan x\right|\\ \frac{\mathsf{expm1}\left(\log t\_0 \cdot 2\right)}{-1 - {t\_0}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (tan x))))
   (/ (expm1 (* (log t_0) 2.0)) (- -1.0 (pow t_0 2.0)))))

C

double code(double x) {
	double t_0 = fabs(tan(x));
	return expm1((log(t_0) * 2.0)) / (-1.0 - pow(t_0, 2.0));
}

Java

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

Python

def code(x):
	t_0 = math.fabs(math.tan(x))
	return math.expm1((math.log(t_0) * 2.0)) / (-1.0 - math.pow(t_0, 2.0))

Julia

function code(x)
	t_0 = abs(tan(x))
	return Float64(expm1(Float64(log(t_0) * 2.0)) / Float64(-1.0 - (t_0 ^ 2.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Tan[x], $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. 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. quot-tan 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. quot-tan 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. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.0

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

3. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. quot-tan N/A

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

   4. lift-tan.f64 N/A

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

   5. quot-tan N/A

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

   6. frac-times N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. mult-flip N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. sqr-sin-a N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. sqr-cos-a N/A

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

   20. lower-+.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-*.f64 99.5

       \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

7. Applied rewrites 49.6%

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

   1. lift-tan.f64 N/A

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

   2. rem-exp-log N/A

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

   3. exp-fabs N/A

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

   4. rem-exp-log N/A

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

   5. lower-fabs.f64 N/A

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

   6. lift-tan.f64 99.3

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

9. Applied rewrites 99.3%

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

    1. lift-tan.f64 N/A

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

    2. rem-exp-log N/A

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

    3. exp-fabs N/A

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

    4. rem-exp-log N/A

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

    5. lower-fabs.f64 N/A

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

    6. lift-tan.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 5: 57.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -0.02:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -0.02)
   (/ -1.0 (- -1.0 (pow (tan x) 2.0)))
   (- (tanh (log (tan x))))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -0.02) {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	} else {
		tmp = -tanh(log(tan(x)));
	}
	return tmp;
}

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) :: tmp
    if (tan(x) <= (-0.02d0)) then
        tmp = (-1.0d0) / ((-1.0d0) - (tan(x) ** 2.0d0))
    else
        tmp = -tanh(log(tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -0.02) {
		tmp = -1.0 / (-1.0 - Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = -Math.tanh(Math.log(Math.tan(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -0.02:
		tmp = -1.0 / (-1.0 - math.pow(math.tan(x), 2.0))
	else:
		tmp = -math.tanh(math.log(math.tan(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -0.02)
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	else
		tmp = Float64(-tanh(log(tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -0.02)
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	else
		tmp = -tanh(log(tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], N[(-1.0 / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Tanh[N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -0.0200000000000000004

   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. quot-tan 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. quot-tan 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. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.0

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. quot-tan N/A

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

      4. lift-tan.f64 N/A

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

      5. quot-tan N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.5

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.7%

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

   if -0.0200000000000000004 < (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

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

      2. lift-tan.f64 N/A

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

      3. quot-tan 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. quot-tan 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. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.0

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. quot-tan N/A

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

      4. lift-tan.f64 N/A

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

      5. quot-tan N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.5

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

   7. Applied rewrites 49.7%

      \[\leadsto \color{blue}{-\tanh \log \tan x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -0.02) 1.0 (- (tanh (log (tan x))))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -0.02) {
		tmp = 1.0;
	} else {
		tmp = -tanh(log(tan(x)));
	}
	return tmp;
}

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) :: tmp
    if (tan(x) <= (-0.02d0)) then
        tmp = 1.0d0
    else
        tmp = -tanh(log(tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -0.02) {
		tmp = 1.0;
	} else {
		tmp = -Math.tanh(Math.log(Math.tan(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -0.02:
		tmp = 1.0
	else:
		tmp = -math.tanh(math.log(math.tan(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -0.02)
		tmp = 1.0;
	else
		tmp = Float64(-tanh(log(tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -0.02)
		tmp = 1.0;
	else
		tmp = -tanh(log(tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], 1.0, (-N[Tanh[N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -0.0200000000000000004

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.3%

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

   if -0.0200000000000000004 < (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

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

      2. lift-tan.f64 N/A

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

      3. quot-tan 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. quot-tan 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. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.0

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}}{1 + \tan x \cdot \tan x} \]

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. quot-tan N/A

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

      4. lift-tan.f64 N/A

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

      5. quot-tan N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. mult-flip N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. sqr-sin-a N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-cos.f64 N/A

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

      23. lower-*.f64 99.5

          \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \color{blue}{\left(2 \cdot x\right)}}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}{1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}} \]

   7. Applied rewrites 49.7%

      \[\leadsto \color{blue}{-\tanh \log \tan x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.001:\\ \;\;\;\;\frac{1 - x \cdot x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.001)
     (/ (- 1.0 (* x x)) (fma x x 1.0))
     1.0)))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= -0.001) {
		tmp = (1.0 - (x * x)) / fma(x, x, 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= -0.001)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / fma(x, x, 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < -1e-3

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.2

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 52.9%

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

   if -1e-3 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.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. Taylor expanded in x around 0

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

   4. Applied rewrites 55.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.5% accurate, 155.8× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

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. Taylor expanded in x around 0

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

4. Applied rewrites 55.3%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))