Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.8s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cos(Float64(x + x))
	t_1 = cos(Float64(x * -2.0))
	return Float64(Float64(1.0 - Float64(fma(t_0, -0.5, 0.5) / fma(0.5, t_0, 0.5))) / Float64(1.0 + Float64(Float64(1.0 - t_1) / Float64(1.0 + t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - N[(N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] / N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + t$95$1), $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 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. distribute-lft-neg-in N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

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

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

   6. distribute-rgt1-in N/A

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

   7. associate-/l/ N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

7. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. frac-times N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sqr-sin-a N/A

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

   14. count-2 N/A

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

   15. lift-cos.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. sqr-cos-a N/A

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

   18. count-2 N/A

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

   19. lower-/.f64 N/A

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

9. Applied rewrites 99.5%

   \[\leadsto \frac{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]
10. 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

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

   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-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. lift-*.f64 N/A

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

   6. sub-neg N/A

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

   7. lift--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(1.0 - N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $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. 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 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. distribute-lft-neg-in N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

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

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

   6. distribute-rgt1-in N/A

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

   7. associate-/l/ N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 63.2%

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

Alternative 5: 59.3% 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(Float64(1.0 - (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[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. 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. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 61.1%

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

Alternative 6: 55.4% 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 57.1%

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

Alternative 7: 55.0% 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 56.7%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))