Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 3.1s

Alternatives: 5

Speedup: 1.2×

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: 99.5% accurate, 0.9× speedup

Math

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

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))))
   (/ (- 1.0 t_1) (+ 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 (1.0 - t_1) / (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(1.0 - t_1) / 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[(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. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. sqr-sin-a N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. sqr-cos-a N/A

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

   18. lower-+.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cos.f64 N/A

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

   21. lower-*.f64 98.9

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

3. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. quot-tan N/A

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

   4. lift-tan.f64 N/A

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

   5. quot-tan N/A

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

   6. frac-times N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. sqr-sin-a N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. sqr-cos-a N/A

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

   18. lower-+.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cos.f64 N/A

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

   21. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f64 99.5

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. count-2-rev N/A

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

   18. lower-+.f64 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f64 99.5

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. count-2-rev N/A

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

   18. lower-+.f64 99.5

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (tan x) (- (tan x)) 1.0) (+ (pow (tan x) 2.0) 1.0)))

C

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

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / Float64((tan(x) ^ 2.0) + 1.0))
end

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lift-tan.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. quot-tan N/A

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

   5. lift-tan.f64 N/A

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

   6. quot-tan N/A

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

   7. frac-times N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. frac-times N/A

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

   15. quot-tan N/A

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

   16. quot-tan N/A

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

   17. pow2 N/A

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

   18. lower-pow.f64 N/A

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

   19. lift-tan.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\color{blue}{{\tan x}^{2} + 1}} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (t_0 - -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
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (t_0 - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $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. lift-tan.f64 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-tan.f64 99.5

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sign-sub-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower--.f64 N/A

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

   17. pow2 N/A

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

   18. lower-pow.f64 N/A

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

   19. lift-tan.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 56.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.3:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.3) 1.0 (/ (- 1.0 (* x x)) (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.3) {
		tmp = 1.0;
	} else {
		tmp = (1.0 - (x * x)) / (1.0 + (x * 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) * tan(x)) <= 1.3d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 - (x * x)) / (1.0d0 + (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 1.3) {
		tmp = 1.0;
	} else {
		tmp = (1.0 - (x * x)) / (1.0 + (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 1.3:
		tmp = 1.0
	else:
		tmp = (1.0 - (x * x)) / (1.0 + (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.3)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 1.3)
		tmp = 1.0;
	else
		tmp = (1.0 - (x * x)) / (1.0 + (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.3], 1.0, N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1.30000000000000004

   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 54.6%

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

   if 1.30000000000000004 < (*.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

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 50.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1 - x \cdot x}{1 + x \cdot \color{blue}{x}} \]
   12. Step-by-step derivation

   13. Applied rewrites 52.3%

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

Alternative 5: 54.6% 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 54.6%

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

Reproduce

herbie shell --seed 2025131 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))