Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.3s

Alternatives: 8

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(1.0 / N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 99.5

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

   5. remove-double-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 60.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := \tan x \cdot \tan x\\ t_2 := 1 + t\_1\\ \mathbf{if}\;\frac{1 - t\_1}{t\_2} \leq 0.25:\\ \;\;\;\;1 + \frac{\frac{-1}{\mathsf{fma}\left(t\_0, -0.5, -0.5\right)}}{\frac{-1}{\mathsf{fma}\left(t\_0, -0.5, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(t\_0 \cdot 0.5 - 0.5\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* (tan x) (tan x))) (t_2 (+ 1.0 t_1)))
   (if (<= (/ (- 1.0 t_1) t_2) 0.25)
     (+ 1.0 (/ (/ -1.0 (fma t_0 -0.5 -0.5)) (/ -1.0 (fma t_0 -0.5 0.5))))
     (/ (+ 1.0 (- (* t_0 0.5) 0.5)) t_2))))

C

double code(double x) {
	double t_0 = cos((x + x));
	double t_1 = tan(x) * tan(x);
	double t_2 = 1.0 + t_1;
	double tmp;
	if (((1.0 - t_1) / t_2) <= 0.25) {
		tmp = 1.0 + ((-1.0 / fma(t_0, -0.5, -0.5)) / (-1.0 / fma(t_0, -0.5, 0.5)));
	} else {
		tmp = (1.0 + ((t_0 * 0.5) - 0.5)) / t_2;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(tan(x) * tan(x))
	t_2 = Float64(1.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 - t_1) / t_2) <= 0.25)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / fma(t_0, -0.5, -0.5)) / Float64(-1.0 / fma(t_0, -0.5, 0.5))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 * 0.5) - 0.5)) / t_2);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 0.25], N[(1.0 + N[(N[(-1.0 / N[(t$95$0 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[(t$95$0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < 0.25

   1. Initial program 98.7%

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

      1. tan-quot N/A

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

      2. lift-tan.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 16.8%

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

   9. Applied rewrites 16.8%

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

   if 0.25 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.7%

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

      1. tan-quot N/A

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

      2. div-inv N/A

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

      3. tan-quot N/A

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

      4. div-inv N/A

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

      5. swap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. inv-pow N/A

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

      15. inv-pow N/A

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

      16. pow-prod-down N/A

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

      17. inv-pow N/A

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

      18. lower-/.f64 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. cos-2 N/A

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

      22. cos-sum N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 77.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \leq 0.25:\\ \;\;\;\;1 + \frac{\frac{-1}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, -0.5\right)}}{\frac{-1}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\cos \left(x + x\right) \cdot 0.5 - 0.5\right)}{1 + \tan x \cdot \tan x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lower--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. 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 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 99.5

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

   5. remove-double-div N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 99.5

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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) / (t_0 - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lower--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. 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-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. neg-mul-1 N/A

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

   13. lift-neg.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-neg.f64 N/A

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

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

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

   17. unpow2 N/A

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

   18. lift-pow.f64 N/A

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

   19. sub-neg N/A

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

   20. lift--.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 6: 59.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot N/A

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

   2. lift-tan.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 59.5%

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

9. Applied rewrites 59.5%

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

Alternative 7: 59.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. tan-quot N/A

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

   2. lift-tan.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 59.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. /-rgt-identity 59.5

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. tan-quot N/A

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

   14. lift-tan.f64 N/A

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

   15. unpow2 N/A

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

   16. lift-pow.f64 59.5

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

9. Applied rewrites 59.5%

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

Alternative 8: 55.1% 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

4. Applied rewrites 55.4%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))