Trigonometry B

Percentage Accurate: 99.5% → 99.7%

Time: 3.8s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

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

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. frac-2neg-rev N/A

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

   13. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

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

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. frac-2neg-rev N/A

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

   13. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. mult-flip-rev N/A

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

    4. lower-/.f64 99.6%

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

    5. lift-fma.f64 N/A

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

    6. *-commutative N/A

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

    7. lift-fma.f64 99.6%

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

11. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.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. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / fma(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] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 6: 98.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/ (- 1.0 (pow (tan x) 2.0)) (/ 1.0 (fma 0.5 (cos (+ x x)) 0.5))))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / (1.0 / fma(0.5, cos((x + x)), 0.5));
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / Float64(1.0 / fma(0.5, cos(Float64(x + x)), 0.5)))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-cos.f64 99.3%

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 98.8%

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

Alternative 7: 79.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.1%

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

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.5%

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

   3. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 99.3%

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 48.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right) \cdot \frac{-1}{\frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. add-flip N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-flip-reverse N/A

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

      18. lower--.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.1%

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

   if -0.01 < (tan.f64 x)

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.5%

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

   3. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-cos.f64 99.3%

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

   8. Applied rewrites 99.3%

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

   10. Applied rewrites 48.9%

       \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. sub-flip-reverse N/A

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

   18. lower--.f64 99.4%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 59.1%

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

Alternative 10: 57.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.05:\\ \;\;\;\;\frac{1 - {x}^{2}}{1 + {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (tan x) (tan x))))
  (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.05)
    (/ (- 1.0 (pow x 2.0)) (+ 1.0 (pow x 2.0)))
    (* (fma 0.5 (cos (+ x x)) 0.5) 1.0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= -0.05) {
		tmp = (1.0 - pow(x, 2.0)) / (1.0 + pow(x, 2.0));
	} else {
		tmp = fma(0.5, cos((x + x)), 0.5) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= -0.05)
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / Float64(1.0 + (x ^ 2.0)));
	else
		tmp = Float64(fma(0.5, cos(Float64(x + x)), 0.5) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * 1.0), $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)))) < -0.050000000000000003

   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

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

      1. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-pow.f64 52.2%

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

   7. Applied rewrites 52.2%

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

   if -0.050000000000000003 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.2%

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

   6. Applied rewrites 55.2%

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

Alternative 11: 57.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.05:\\ \;\;\;\;\frac{1 - {x}^{2}}{1 + {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (tan x) (tan x))))
  (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.05)
    (/ (- 1.0 (pow x 2.0)) (+ 1.0 (pow x 2.0)))
    (/ 1.0 (/ 1.0 (fma 0.5 (cos (+ x x)) 0.5))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= -0.05) {
		tmp = (1.0 - pow(x, 2.0)) / (1.0 + pow(x, 2.0));
	} else {
		tmp = 1.0 / (1.0 / fma(0.5, cos((x + x)), 0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= -0.05)
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / Float64(1.0 + (x ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(0.5, cos(Float64(x + x)), 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $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)))) < -0.050000000000000003

   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

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

      1. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-pow.f64 52.2%

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

   7. Applied rewrites 52.2%

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

   if -0.050000000000000003 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-cos.f64 N/A

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

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

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

      17. count-2 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   6. Applied rewrites 55.2%

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

Alternative 12: 57.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.05:\\ \;\;\;\;\frac{1 - {x}^{2}}{1 + {x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, 1, 0.5\right)}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (tan x) (tan x))))
  (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.05)
    (/ (- 1.0 (pow x 2.0)) (+ 1.0 (pow x 2.0)))
    (/ 1.0 (/ 1.0 (fma 0.5 1.0 0.5))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= -0.05) {
		tmp = (1.0 - pow(x, 2.0)) / (1.0 + pow(x, 2.0));
	} else {
		tmp = 1.0 / (1.0 / fma(0.5, 1.0, 0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= -0.05)
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / Float64(1.0 + (x ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(0.5, 1.0, 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(0.5 * 1.0 + 0.5), $MachinePrecision]), $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)))) < -0.050000000000000003

   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

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

      1. lower-pow.f64 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-pow.f64 52.2%

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

   7. Applied rewrites 52.2%

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

   if -0.050000000000000003 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 55.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lift-cos.f64 N/A

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

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

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

      17. count-2 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-+.f64 N/A

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

   6. Applied rewrites 55.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 54.8%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.5, \color{blue}{1}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (- (tan (fabs x)) -1.0)
 (* (fma -0.5 (cos (+ (fabs x) (fabs x))) -0.5) -1.0)))

C

double code(double x) {
	return (tan(fabs(x)) - -1.0) * (fma(-0.5, cos((fabs(x) + fabs(x))), -0.5) * -1.0);
}

Julia

function code(x)
	return Float64(Float64(tan(abs(x)) - -1.0) * Float64(fma(-0.5, cos(Float64(abs(x) + abs(x))), -0.5) * -1.0))
end

Wolfram

code[x_] := N[(N[(N[Tan[N[Abs[x], $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(-0.5 * N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -0.5), $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

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

   2. frac-2neg N/A

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. add-flip N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. sub-flip-reverse N/A

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

   18. lower--.f64 99.4%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 56.1%

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

8. Applied rewrites 56.1%

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

Alternative 14: 54.8% accurate, 12.1× speedup

Math

\[\frac{1}{\frac{1}{\mathsf{fma}\left(0.5, 1, 0.5\right)}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 1.0 (/ 1.0 (fma 0.5 1.0 0.5))))

C

double code(double x) {
	return 1.0 / (1.0 / fma(0.5, 1.0, 0.5));
}

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 / fma(0.5, 1.0, 0.5)))
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 / N[(0.5 * 1.0 + 0.5), $MachinePrecision]), $MachinePrecision]), $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

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

4. Applied rewrites 55.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. frac-times N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. lift-cos.f64 N/A

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

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

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

   17. count-2 N/A

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

   18. lift-+.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. lift-+.f64 N/A

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

6. Applied rewrites 55.2%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 54.8%

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

Alternative 15: 50.6% accurate, 18.3× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (fma (* x x) -2.0 1.0))

C

double code(double x) {
	return fma((x * x), -2.0, 1.0);
}

Julia

function code(x)
	return fma(Float64(x * x), -2.0, 1.0)
end

Wolfram

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

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 50.6%

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

4. Applied rewrites 50.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

      \[\leadsto -2 \cdot {x}^{2} + 1 \]

   4. *-commutative N/A

      \[\leadsto {x}^{2} \cdot -2 + 1 \]

   5. lower-fma.f64 50.6%

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 50.6%

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

6. Applied rewrites 50.6%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{-2}, 1\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))