Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 3.2s

Alternatives: 11

Speedup: 0.9×

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. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. 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. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. tan-quot 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. sqr-sin-a-rev N/A

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

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

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f64 N/A

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

   16. +-commutative N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-cos.f64 N/A

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

   12. count-2-rev N/A

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 2: 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]]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{{\tan x}^{2} - -1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. negate-sub-reverse N/A

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

   8. lower--.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-tan.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 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

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. negate-sub-reverse N/A

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

   8. lower--.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-tan.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. pow2 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-pow.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 5: 60.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -0.02:\\ \;\;\;\;\frac{1 - \tan x \cdot \tan x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1 - {\tan x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -0.02)
   (/ (- 1.0 (* (tan x) (tan x))) 1.0)
   (/ (expm1 (* (log (tan x)) 2.0)) (- -1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -0.02) {
		tmp = (1.0 - (tan(x) * tan(x))) / 1.0;
	} else {
		tmp = expm1((log(tan(x)) * 2.0)) / (-1.0 - pow(tan(x), 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -0.02) {
		tmp = (1.0 - (Math.tan(x) * Math.tan(x))) / 1.0;
	} else {
		tmp = Math.expm1((Math.log(Math.tan(x)) * 2.0)) / (-1.0 - Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -0.02:
		tmp = (1.0 - (math.tan(x) * math.tan(x))) / 1.0
	else:
		tmp = math.expm1((math.log(math.tan(x)) * 2.0)) / (-1.0 - math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -0.02)
		tmp = Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / 1.0);
	else
		tmp = Float64(expm1(Float64(log(tan(x)) * 2.0)) / Float64(-1.0 - (tan(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(Exp[N[(N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 19.7%

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

   if -0.0200000000000000004 < (tan.f64 x)

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub-reverse N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-tan.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Applied rewrites 66.4%

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

Alternative 6: 59.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := 1 - t\_0\\ \mathbf{if}\;\frac{t\_1}{1 + t\_0} \leq 0.22:\\ \;\;\;\;\frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \cos \left(x + x\right) \cdot 0.5}{1}}{{\tan x}^{2} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (- 1.0 t_0)))
   (if (<= (/ t_1 (+ 1.0 t_0)) 0.22)
     (/ t_1 1.0)
     (/
      (- 1.0 (/ (- 0.5 (* (cos (+ x x)) 0.5)) 1.0))
      (- (pow (tan x) 2.0) -1.0)))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((t_1 / (1.0 + t_0)) <= 0.22) {
		tmp = t_1 / 1.0;
	} else {
		tmp = (1.0 - ((0.5 - (cos((x + x)) * 0.5)) / 1.0)) / (pow(tan(x), 2.0) - -1.0);
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    t_1 = 1.0d0 - t_0
    if ((t_1 / (1.0d0 + t_0)) <= 0.22d0) then
        tmp = t_1 / 1.0d0
    else
        tmp = (1.0d0 - ((0.5d0 - (cos((x + x)) * 0.5d0)) / 1.0d0)) / ((tan(x) ** 2.0d0) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((t_1 / (1.0 + t_0)) <= 0.22) {
		tmp = t_1 / 1.0;
	} else {
		tmp = (1.0 - ((0.5 - (Math.cos((x + x)) * 0.5)) / 1.0)) / (Math.pow(Math.tan(x), 2.0) - -1.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	t_1 = 1.0 - t_0
	tmp = 0
	if (t_1 / (1.0 + t_0)) <= 0.22:
		tmp = t_1 / 1.0
	else:
		tmp = (1.0 - ((0.5 - (math.cos((x + x)) * 0.5)) / 1.0)) / (math.pow(math.tan(x), 2.0) - -1.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (Float64(t_1 / Float64(1.0 + t_0)) <= 0.22)
		tmp = Float64(t_1 / 1.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) / 1.0)) / Float64((tan(x) ^ 2.0) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if ((t_1 / (1.0 + t_0)) <= 0.22)
		tmp = t_1 / 1.0;
	else
		tmp = (1.0 - ((0.5 - (cos((x + x)) * 0.5)) / 1.0)) / ((tan(x) ^ 2.0) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.22], N[(t$95$1 / 1.0), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]

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)))) < 0.220000000000000001

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 16.6%

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

   if 0.220000000000000001 < (/.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 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub-reverse N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-tan.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. tan-quot N/A

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

      6. frac-times N/A

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

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

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

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

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.8%

      \[\leadsto \frac{1 - \frac{0.5 - \cos \left(x + x\right) \cdot 0.5}{\color{blue}{1}}}{{\tan x}^{2} - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. 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. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. tan-quot 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. sqr-sin-a-rev N/A

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

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

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f64 N/A

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

   16. +-commutative N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-cos.f64 N/A

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

   12. count-2-rev N/A

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 59.0%

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

Alternative 8: 57.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{1} \end{array} \]

FPCore

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

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / 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 - (tan(x) * tan(x))) / 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

4. Applied rewrites 59.0%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{1}} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{{\tan x}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) 4e-5)
   (/ 1.0 (- (pow (tan x) 2.0) -1.0))
   (/ (expm1 (* (log (tan x)) 2.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (tan(x) <= 4e-5) {
		tmp = 1.0 / (pow(tan(x), 2.0) - -1.0);
	} else {
		tmp = expm1((log(tan(x)) * 2.0)) / -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= 4e-5) {
		tmp = 1.0 / (Math.pow(Math.tan(x), 2.0) - -1.0);
	} else {
		tmp = Math.expm1((Math.log(Math.tan(x)) * 2.0)) / -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= 4e-5:
		tmp = 1.0 / (math.pow(math.tan(x), 2.0) - -1.0)
	else:
		tmp = math.expm1((math.log(math.tan(x)) * 2.0)) / -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= 4e-5)
		tmp = Float64(1.0 / Float64((tan(x) ^ 2.0) - -1.0));
	else
		tmp = Float64(expm1(Float64(log(tan(x)) * 2.0)) / -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], 4e-5], N[(1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[(N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < 4.00000000000000033e-5

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub-reverse N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-tan.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

      1. tan-quot 69.9

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

      2. tan-quot 69.9

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

      3. frac-times 69.9

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

      4. sqr-sin-a-rev 69.9

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

      5. sqr-cos-a-rev 69.9

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

   6. Applied rewrites 69.9%

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

   if 4.00000000000000033e-5 < (tan.f64 x)

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub-reverse N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-tan.f64 99.0

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

   3. Applied rewrites 99.0%

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 20.0%

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

Alternative 10: 54.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -0.02) 1.0 (/ (expm1 (* (log (tan x)) 2.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -0.02) {
		tmp = 1.0;
	} else {
		tmp = expm1((log(tan(x)) * 2.0)) / -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -0.02) {
		tmp = 1.0;
	} else {
		tmp = Math.expm1((Math.log(Math.tan(x)) * 2.0)) / -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -0.02:
		tmp = 1.0
	else:
		tmp = math.expm1((math.log(math.tan(x)) * 2.0)) / -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -0.02)
		tmp = 1.0;
	else
		tmp = Float64(expm1(Float64(log(tan(x)) * 2.0)) / -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], 1.0, N[(N[(Exp[N[(N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]] - 1), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 11.2%

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

   if -0.0200000000000000004 < (tan.f64 x)

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. negate-sub-reverse N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-tan.f64 99.6

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

   3. Applied rewrites 99.6%

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.5%

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

Alternative 11: 32.4% 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 54.7%

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

Reproduce

herbie shell --seed 2025116 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))