Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.2s

Alternatives: 11

Speedup: 1.0×

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

real(8) function code(x)
    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

real(8) function code(x)
    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

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = cos((x * (-2.0d0)))
    t_1 = 1.0d0 - t_0
    code = (1.0d0 + (t_1 / ((-1.0d0) - t_0))) / (1.0d0 + (t_1 / (1.0d0 + t_0)))
end function

Java

public static double code(double x) {
	double t_0 = Math.cos((x * -2.0));
	double t_1 = 1.0 - t_0;
	return (1.0 + (t_1 / (-1.0 - t_0))) / (1.0 + (t_1 / (1.0 + t_0)));
}

Python

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

Julia

function code(x)
	t_0 = cos(Float64(x * -2.0))
	t_1 = Float64(1.0 - t_0)
	return Float64(Float64(1.0 + Float64(t_1 / Float64(-1.0 - t_0))) / Float64(1.0 + Float64(t_1 / Float64(1.0 + t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = cos((x * -2.0));
	t_1 = 1.0 - t_0;
	tmp = (1.0 + (t_1 / (-1.0 - t_0))) / (1.0 + (t_1 / (1.0 + t_0)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. lower-+.f64 N/A

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

   12. cos-2 N/A

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

   13. cos-sum N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. sqr-sin-a N/A

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

   18. lower--.f64 N/A

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

   19. cos-2 N/A

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

   20. cos-sum N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. sqr-sin-a N/A

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

   8. count-2 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sqr-cos-a N/A

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

   14. count-2 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f64 N/A

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

   14. clear-num N/A

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

   15. lift-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

4. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

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

real(8) function code(x)
    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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. sqr-sin-a N/A

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

   8. count-2 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sqr-cos-a N/A

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

   14. count-2 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. clear-num N/A

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

   20. lift-/.f64 N/A

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.5%

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

Alternative 4: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{1}{0.5 - 0.5 \cdot \cos \left(x + x\right)}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(1.0 + Float64(1.0 / Float64(1.0 / Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. lower-+.f64 N/A

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

   12. cos-2 N/A

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

   13. cos-sum N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. sqr-sin-a N/A

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

   18. lower--.f64 N/A

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

   19. cos-2 N/A

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

   20. cos-sum N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 62.2%

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

Alternative 5: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\\ \left(1 - {\tan x}^{2}\right) \cdot \frac{1}{1 + \frac{1}{t\_0 \cdot t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* x x)
           (fma
            x
            (* x (fma (* x x) -0.0021164021164021165 -0.022222222222222223))
            -0.3333333333333333)
           1.0)
          x)))
   (* (- 1.0 (pow (tan x) 2.0)) (/ 1.0 (+ 1.0 (/ 1.0 (* t_0 t_0)))))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(x, (x * fma((x * x), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / x;
	return (1.0 - pow(tan(x), 2.0)) * (1.0 / (1.0 + (1.0 / (t_0 * t_0))));
}

Julia

function code(x)
	t_0 = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) * Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(t_0 * t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0021164021164021165 + -0.022222222222222223), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}}{x}\right)}^{-2}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)}{x}\right)}^{-2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \color{blue}{\frac{-1}{3}}, 1\right)}{x}\right)}^{-2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right)}, 1\right)}{x}\right)}^{-2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   11. sub-neg N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-2}{945}} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-2}{945} + \color{blue}{\frac{-1}{45}}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945}, \frac{-1}{45}\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   16. lower-*.f64 60.7

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 60.7%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}}^{-2}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}}{1 - \tan x \cdot \tan x}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \cdot \left(1 - \tan x \cdot \tan x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \cdot \left(1 - \tan x \cdot \tan x\right)} \]

9. Applied rewrites 60.7%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}}} \cdot \left(1 - {\tan x}^{2}\right)} \]

10. Final simplification 60.7%

    \[\leadsto \left(1 - {\tan x}^{2}\right) \cdot \frac{1}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}}} \]
11. Add Preprocessing

Alternative 6: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\\ \frac{1 - {\tan x}^{2}}{1 + \frac{1}{t\_0 \cdot t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* x x)
           (fma
            x
            (* x (fma (* x x) -0.0021164021164021165 -0.022222222222222223))
            -0.3333333333333333)
           1.0)
          x)))
   (/ (- 1.0 (pow (tan x) 2.0)) (+ 1.0 (/ 1.0 (* t_0 t_0))))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(x, (x * fma((x * x), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / x;
	return (1.0 - pow(tan(x), 2.0)) / (1.0 + (1.0 / (t_0 * t_0)));
}

Julia

function code(x)
	t_0 = Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / Float64(1.0 + Float64(1.0 / Float64(t_0 * t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.0021164021164021165 + -0.022222222222222223), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]}, N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}}{x}\right)}^{-2}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)}{x}\right)}^{-2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \color{blue}{\frac{-1}{3}}, 1\right)}{x}\right)}^{-2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right)}, 1\right)}{x}\right)}^{-2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   11. sub-neg N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-2}{945}} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-2}{945} + \color{blue}{\frac{-1}{45}}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945}, \frac{-1}{45}\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   16. lower-*.f64 60.7

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 60.7%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}}^{-2}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\tan x \cdot \tan x}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   2. pow2 N/A

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   3. lower-pow.f64 60.7

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \]

   6. pow-sqr N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-1} \cdot {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-1}}} \]

   7. pow-prod-down N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-1}}} \]

   8. unpow-1 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}}}} \]

9. Applied rewrites 60.7%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}}}} \]
10. Add Preprocessing

Alternative 7: 60.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\mathsf{fma}\left(x, x \cdot 0.06666666666666667, \mathsf{fma}\left(1, \frac{1}{x \cdot x}, -0.6666666666666666\right)\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (* (tan x) (tan x)))
  (+
   1.0
   (/
    1.0
    (fma
     x
     (* x 0.06666666666666667)
     (fma 1.0 (/ 1.0 (* x x)) -0.6666666666666666))))))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (1.0 / fma(x, (x * 0.06666666666666667), fma(1.0, (1.0 / (x * x)), -0.6666666666666666))));
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(1.0 + Float64(1.0 / fma(x, Float64(x * 0.06666666666666667), fma(1.0, Float64(1.0 / Float64(x * x)), -0.6666666666666666)))))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 / N[(x * N[(x * 0.06666666666666667), $MachinePrecision] + N[(1.0 * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. lower-+.f64 N/A

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

   12. cos-2 N/A

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

   13. cos-sum N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. sqr-sin-a N/A

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

   18. lower--.f64 N/A

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

   19. cos-2 N/A

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

   20. cos-sum N/A

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

   21. lower-*.f64 N/A

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

   22. lower-cos.f64 N/A

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

   23. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{{x}^{2}}}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{{x}^{2}}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right) + 1}}{{x}^{2}}}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{15} \cdot {x}^{2} - \frac{2}{3}, 1\right)}}{{x}^{2}}}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{15} \cdot {x}^{2} - \frac{2}{3}, 1\right)}{{x}^{2}}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{15} \cdot {x}^{2} - \frac{2}{3}, 1\right)}{{x}^{2}}}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 1\right)}{{x}^{2}}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{15} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 1\right)}{{x}^{2}}}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{15} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 1\right)}{{x}^{2}}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{15} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 1\right)}{{x}^{2}}}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{15} \cdot x\right) + \color{blue}{\frac{-2}{3}}, 1\right)}{{x}^{2}}}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{15} \cdot x, \frac{-2}{3}\right)}, 1\right)}{{x}^{2}}}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{15}}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{15}}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

   14. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{15}, \frac{-2}{3}\right), 1\right)}{\color{blue}{x \cdot x}}}} \]

   15. lower-*.f64 55.2

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.06666666666666667, -0.6666666666666666\right), 1\right)}{\color{blue}{x \cdot x}}}} \]

7. Applied rewrites 55.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.06666666666666667, -0.6666666666666666\right), 1\right)}{x \cdot x}}}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{{x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{15} + \frac{1}{{x}^{4}}\right) - \frac{2}{3} \cdot \frac{1}{{x}^{2}}\right)}}} \]
9. Step-by-step derivation

10. Applied rewrites 60.6%

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.06666666666666667}, \mathsf{fma}\left(1, \frac{1}{x \cdot x}, -0.6666666666666666\right)\right)}} \]
11. Add Preprocessing

Alternative 8: 59.9% 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

real(8) function code(x)
    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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 60.3%

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

Alternative 9: 56.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / 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. Add Preprocessing
3. 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-neg 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

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

4. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 56.8%

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

Alternative 10: 55.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   1.0
   (pow
    (/
     (fma
      (* x x)
      (fma
       (* x x)
       (fma (* x x) -0.0021164021164021165 -0.022222222222222223)
       -0.3333333333333333)
      1.0)
     x)
    -2.0))))

C

double code(double x) {
	return 1.0 / (1.0 + pow((fma((x * x), fma((x * x), fma((x * x), -0.0021164021164021165, -0.022222222222222223), -0.3333333333333333), 1.0) / x), -2.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + (Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.0021164021164021165, -0.022222222222222223), -0.3333333333333333), 1.0) / x) ^ -2.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[Power[N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0021164021164021165 + -0.022222222222222223), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}}{x}\right)}^{-2}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{x}\right)}^{-2}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)}{x}\right)}^{-2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \color{blue}{\frac{-1}{3}}, 1\right)}{x}\right)}^{-2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right)}, 1\right)}{x}\right)}^{-2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   11. sub-neg N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-2}{945}} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-2}{945} + \color{blue}{\frac{-1}{45}}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-2}{945}, \frac{-1}{45}\right)}, \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]

   16. lower-*.f64 60.7

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 60.7%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}}^{-2}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-2}{945}, \frac{-1}{45}\right), \frac{-1}{3}\right), 1\right)}{x}\right)}^{-2}} \]
9. Step-by-step derivation

10. Applied rewrites 56.4%

    \[\leadsto \frac{\color{blue}{1}}{1 + {\left(\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{x}\right)}^{-2}} \]
11. Add Preprocessing

Alternative 11: 55.8% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.4%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))