Trigonometry B

Percentage Accurate: 99.5% → 99.4%

Time: 9.0s

Alternatives: 9

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(N[(N[Tan[x], $MachinePrecision] * N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[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 egg-rr 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. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-*.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.4

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

6. Applied egg-rr 99.4%

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

   1. tan-quot N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-/.f64 99.5

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\tan x, 1\right)}{\mathsf{fma}\left(\tan x \cdot \frac{1}{\cos x}, \sin x, 1\right)} \]
10. Add Preprocessing

Alternative 2: 60.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{1 - t\_0}{t\_1} \leq 0.25:\\ \;\;\;\;1 - {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(0.5 \cdot \cos \left(x + x\right) - 0.5\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ (- 1.0 t_0) t_1) 0.25)
     (- 1.0 (pow (tan x) 2.0))
     (/ (+ 1.0 (- (* 0.5 (cos (+ x x))) 0.5)) t_1))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = 1.0 + t_0;
	double tmp;
	if (((1.0 - t_0) / t_1) <= 0.25) {
		tmp = 1.0 - pow(tan(x), 2.0);
	} else {
		tmp = (1.0 + ((0.5 * cos((x + x))) - 0.5)) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    t_1 = 1.0d0 + t_0
    if (((1.0d0 - t_0) / t_1) <= 0.25d0) then
        tmp = 1.0d0 - (tan(x) ** 2.0d0)
    else
        tmp = (1.0d0 + ((0.5d0 * cos((x + x))) - 0.5d0)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double t_1 = 1.0 + t_0;
	double tmp;
	if (((1.0 - t_0) / t_1) <= 0.25) {
		tmp = 1.0 - Math.pow(Math.tan(x), 2.0);
	} else {
		tmp = (1.0 + ((0.5 * Math.cos((x + x))) - 0.5)) / t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	t_1 = 1.0 + t_0
	tmp = 0
	if ((1.0 - t_0) / t_1) <= 0.25:
		tmp = 1.0 - math.pow(math.tan(x), 2.0)
	else:
		tmp = (1.0 + ((0.5 * math.cos((x + x))) - 0.5)) / t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / t_1) <= 0.25)
		tmp = Float64(1.0 - (tan(x) ^ 2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(0.5 * cos(Float64(x + x))) - 0.5)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	t_1 = 1.0 + t_0;
	tmp = 0.0;
	if (((1.0 - t_0) / t_1) <= 0.25)
		tmp = 1.0 - (tan(x) ^ 2.0);
	else
		tmp = (1.0 + ((0.5 * cos((x + x))) - 0.5)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], 0.25], N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / t$95$1), $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 egg-rr 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. Simplified 16.8%

      \[\leadsto \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{\color{blue}{1}} \]
   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-cos.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-tan.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift--.f64 16.8

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

   9. Applied egg-rr 16.8%

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

   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. tan-quot N/A

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

      3. frac-times N/A

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

      4. lower-/.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. sqr-cos-a N/A

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

      13. lower-+.f64 N/A

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

      14. cos-2 N/A

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

      15. cos-sum N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 77.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \leq 0.25:\\ \;\;\;\;1 - {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(0.5 \cdot \cos \left(x + x\right) - 0.5\right)}{1 + \tan x \cdot \tan x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

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 egg-rr 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 egg-rr 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 4: 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 egg-rr 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 egg-rr 99.5%

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

Alternative 5: 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

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. 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 egg-rr 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-tan.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

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

   3. pow2 N/A

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

   4. lift-pow.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 6: 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-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 egg-rr 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 egg-rr 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. lift-fma.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. remove-double-div N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. lift-/.f64 N/A

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

8. Applied egg-rr 99.5%

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

Alternative 7: 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 egg-rr 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. Simplified 59.5%

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

9. Applied egg-rr 59.5%

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

10. Final simplification 59.5%

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

Alternative 8: 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 egg-rr 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. Simplified 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-cos.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. tan-quot N/A

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

   8. lift-tan.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift--.f64 59.5

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

9. Applied egg-rr 59.5%

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

10. Final simplification 59.5%

    \[\leadsto 1 - {\tan x}^{2} \]
11. Add Preprocessing

Alternative 9: 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 egg-rr 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)))))