Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.7s

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

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

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 2: 60.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t\_0 \leq 0.6:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \frac{1}{x \cdot \left(x \cdot \left(0.010582010582010581 \cdot \left(x \cdot x\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= t_0 0.6)
     (/ (- 1.0 (/ (fma (cos (+ x x)) -0.5 0.5) 1.0)) (fma (tan x) (tan x) 1.0))
     (/
      (- 1.0 t_0)
      (+ 1.0 (/ 1.0 (* x (* x (* 0.010582010582010581 (* x x))))))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (1.0 - (fma(cos((x + x)), -0.5, 0.5) / 1.0)) / fma(tan(x), tan(x), 1.0);
	} else {
		tmp = (1.0 - t_0) / (1.0 + (1.0 / (x * (x * (0.010582010582010581 * (x * x))))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (t_0 <= 0.6)
		tmp = Float64(Float64(1.0 - Float64(fma(cos(Float64(x + x)), -0.5, 0.5) / 1.0)) / fma(tan(x), tan(x), 1.0));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + Float64(1.0 / Float64(x * Float64(x * Float64(0.010582010582010581 * Float64(x * x)))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6], N[(N[(1.0 - N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[(1.0 / N[(x * N[(x * N[(0.010582010582010581 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 0.599999999999999978

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. frac-times N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sqr-sin-a N/A

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

      14. count-2 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. sqr-cos-a N/A

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

      22. count-2 N/A

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

      23. lift-+.f64 N/A

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

      24. lift-cos.f64 N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 81.4%

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

   if 0.599999999999999978 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 98.8%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{{x}^{2}}}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{{x}^{2}}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right) + 1}}{{x}^{2}}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}, 1\right)}}{{x}^{2}}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}, 1\right)}{{x}^{2}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}, 1\right)}{{x}^{2}}}} \]

      6. sub-neg N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 1\right)}{{x}^{2}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) + \color{blue}{\frac{-2}{3}}, 1\right)}{{x}^{2}}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{15} + \frac{2}{189} \cdot {x}^{2}, \frac{-2}{3}\right)}, 1\right)}{{x}^{2}}}} \]

      9. unpow2 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{15} + \frac{2}{189} \cdot {x}^{2}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{15} + \frac{2}{189} \cdot {x}^{2}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{189} \cdot {x}^{2} + \frac{1}{15}}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{2}{189}} + \frac{1}{15}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{189}, \frac{1}{15}\right)}, \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      14. unpow2 N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{189}, \frac{1}{15}\right), \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{189}, \frac{1}{15}\right), \frac{-2}{3}\right), 1\right)}{{x}^{2}}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{2}{189}, \frac{1}{15}\right), \frac{-2}{3}\right), 1\right)}{\color{blue}{x \cdot x}}}} \]

      17. lower-*.f64 6.2

          \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.010582010582010581, 0.06666666666666667\right), -0.6666666666666666\right), 1\right)}{\color{blue}{x \cdot x}}}} \]

   7. Applied rewrites 6.2%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.010582010582010581, 0.06666666666666667\right), -0.6666666666666666\right), 1\right)}{x \cdot x}}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{2}{189} \cdot \color{blue}{{x}^{4}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.7%

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot 0.010582010582010581\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{x \cdot \left(x \cdot \left(0.010582010582010581 \cdot \left(x \cdot x\right)\right)\right)}}\\ \end{array} \]
5. 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, 1.2× 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 99.0

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

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 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 63.2%

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

Alternative 5: 59.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. frac-times N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. sqr-sin-a N/A

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

   14. count-2 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift--.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. lift-cos.f64 N/A

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

   21. sqr-cos-a N/A

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

   22. count-2 N/A

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

   23. lift-+.f64 N/A

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

   24. lift-cos.f64 N/A

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

   25. lift-*.f64 N/A

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.1%

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

Alternative 6: 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 7: 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 8: 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)))))