Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Alternative 1: 99.5% accurate, 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 99.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   5. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

4. Applied egg-rr 99.4%

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

Alternative 2: 60.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;e^{-2 \cdot \mathsf{log1p}\left(t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (if (<= (* (tan x) (tan x)) 0.6) (exp (* -2.0 (log1p t_0))) (- 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	double tmp;
	if ((tan(x) * tan(x)) <= 0.6) {
		tmp = exp((-2.0 * log1p(t_0)));
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 0.6) {
		tmp = Math.exp((-2.0 * Math.log1p(t_0)));
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 0.6:
		tmp = math.exp((-2.0 * math.log1p(t_0)))
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 0.6)
		tmp = exp(Float64(-2.0 * log1p(t_0)));
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 0.6], N[Exp[N[(-2.0 * N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - t$95$0), $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. div-inv N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right), \color{blue}{\left(\frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}\right)}\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left(1 + {\tan x}^{2}\right)}^{-2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), -2\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.5%

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

      1. *-lft-identity N/A

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

      2. pow-to-exp N/A

         \[\leadsto e^{\log \left(1 + {\tan x}^{2}\right) \cdot -2} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(1 + {\tan x}^{2}\right) \cdot -2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-2 \cdot \log \left(1 + {\tan x}^{2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \log \left(1 + {\tan x}^{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \log \left(1 + {\tan x}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)\right)\right) \]

      7. pow-flip N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \log \left(1 + \frac{1}{{\tan x}^{-2}}\right)\right)\right) \]

      8. log1p-define N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \left(\mathsf{log1p}\left(\frac{1}{{\tan x}^{-2}}\right)\right)\right)\right) \]

      9. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{log1p.f64}\left(\left(\frac{1}{{\tan x}^{-2}}\right)\right)\right)\right) \]

      10. pow-flip N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{log1p.f64}\left(\left({\tan x}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{log1p.f64}\left(\left({\tan x}^{2}\right)\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{log1p.f64}\left(\mathsf{pow.f64}\left(\tan x, 2\right)\right)\right)\right) \]

      13. tan-lowering-tan.f64 79.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{log1p.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right)\right) \]

   9. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{e^{-2 \cdot \mathsf{log1p}\left({\tan x}^{2}\right)}} \]

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

   1. Initial program 98.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      5. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

      9. tan-lowering-tan.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 16.9%

      \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;e^{-2 \cdot \mathsf{log1p}\left({\tan x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(x \cdot 2\right)\\ \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + \frac{0.5 - t\_0}{0.5 + t\_0}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* x 2.0)))))
   (if (<= (* (tan x) (tan x)) 0.6)
     (pow (+ 1.0 (/ (- 0.5 t_0) (+ 0.5 t_0))) -2.0)
     (- 1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	double t_0 = 0.5 * cos((x * 2.0));
	double tmp;
	if ((tan(x) * tan(x)) <= 0.6) {
		tmp = pow((1.0 + ((0.5 - t_0) / (0.5 + t_0))), -2.0);
	} else {
		tmp = 1.0 - pow(tan(x), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos((x * 2.0d0))
    if ((tan(x) * tan(x)) <= 0.6d0) then
        tmp = (1.0d0 + ((0.5d0 - t_0) / (0.5d0 + t_0))) ** (-2.0d0)
    else
        tmp = 1.0d0 - (tan(x) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.5 * Math.cos((x * 2.0));
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 0.6) {
		tmp = Math.pow((1.0 + ((0.5 - t_0) / (0.5 + t_0))), -2.0);
	} else {
		tmp = 1.0 - Math.pow(Math.tan(x), 2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 * math.cos((x * 2.0))
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 0.6:
		tmp = math.pow((1.0 + ((0.5 - t_0) / (0.5 + t_0))), -2.0)
	else:
		tmp = 1.0 - math.pow(math.tan(x), 2.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 * cos(Float64(x * 2.0)))
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 0.6)
		tmp = Float64(1.0 + Float64(Float64(0.5 - t_0) / Float64(0.5 + t_0))) ^ -2.0;
	else
		tmp = Float64(1.0 - (tan(x) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 * cos((x * 2.0));
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 0.6)
		tmp = (1.0 + ((0.5 - t_0) / (0.5 + t_0))) ^ -2.0;
	else
		tmp = 1.0 - (tan(x) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 0.6], N[Power[N[(1.0 + N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. div-inv N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right), \color{blue}{\left(\frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}\right)}\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left(1 + {\tan x}^{2}\right)}^{-2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), -2\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.5%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), -2\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\sin x}{\cos x} \cdot \tan x\right)\right), -2\right)\right) \]

      3. tan-quot N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right), -2\right)\right) \]

      4. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \cos x}\right)\right), -2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{{\sin x}^{2}}{\cos x \cdot \cos x}\right)\right), -2\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{{\sin x}^{\left(\frac{4}{2}\right)}}{\cos x \cdot \cos x}\right)\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{{\sin x}^{\left(\frac{4}{2}\right)}}{{\cos x}^{2}}\right)\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{{\sin x}^{\left(\frac{4}{2}\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}\right)\right), -2\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({\sin x}^{\left(\frac{4}{2}\right)}\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({\sin x}^{2}\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\sin x \cdot \sin x\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      12. sqr-sin-a N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(\frac{1}{2}\right), \cos \left(2 \cdot x\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(2 \cdot x\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      19. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot x\right)\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right)\right)\right), \left({\cos x}^{\left(\frac{4}{2}\right)}\right)\right)\right), -2\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right)\right)\right), \left({\cos x}^{2}\right)\right)\right), -2\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right)\right)\right), \left(\cos x \cdot \cos x\right)\right)\right), -2\right)\right) \]

      23. sqr-cos-a N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)\right), -2\right)\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right)\right)\right), \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)\right), -2\right)\right) \]

   9. Applied egg-rr 79.5%

      \[\leadsto 1 \cdot {\left(1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot x\right)}}\right)}^{-2} \]

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

   1. Initial program 98.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      5. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

      9. tan-lowering-tan.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 16.9%

      \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + \frac{0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)}{0.5 + 0.5 \cdot \cos \left(x \cdot 2\right)}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + t\_0\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (if (<= (* (tan x) (tan x)) 0.6) (pow (+ 1.0 t_0) -2.0) (- 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	double tmp;
	if ((tan(x) * tan(x)) <= 0.6) {
		tmp = pow((1.0 + t_0), -2.0);
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) ** 2.0d0
    if ((tan(x) * tan(x)) <= 0.6d0) then
        tmp = (1.0d0 + t_0) ** (-2.0d0)
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 0.6) {
		tmp = Math.pow((1.0 + t_0), -2.0);
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 0.6:
		tmp = math.pow((1.0 + t_0), -2.0)
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 0.6)
		tmp = Float64(1.0 + t_0) ^ -2.0;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 0.6)
		tmp = (1.0 + t_0) ^ -2.0;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 0.6], N[Power[N[(1.0 + t$95$0), $MachinePrecision], -2.0], $MachinePrecision], N[(1.0 - t$95$0), $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. div-inv N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right), \color{blue}{\left(\frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}\right)}\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left(1 + {\tan x}^{2}\right)}^{-2}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), -2\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 79.5%

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 + \frac{1}{{\tan x}^{-2}}\right), \color{blue}{-2}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{{\tan x}^{-2}}\right)\right), -2\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left({\tan x}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)\right), -2\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left({\tan x}^{2}\right)\right), -2\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), -2\right) \]

      9. tan-lowering-tan.f64 79.5%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), -2\right) \]

   9. Applied egg-rr 79.5%

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

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

   1. Initial program 98.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      5. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

      9. tan-lowering-tan.f64 98.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 16.9%

      \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + {\tan x}^{2}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - {\tan x}^{4}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (- 1.0 (pow (tan x) 4.0))))

C

double code(double x) {
	return 1.0 / (1.0 - pow(tan(x), 4.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 - (tan(x) ** 4.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.pow(Math.tan(x), 4.0));
}

Python

def code(x):
	return 1.0 / (1.0 - math.pow(math.tan(x), 4.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 - (tan(x) ^ 4.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - (tan(x) ^ 4.0));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. flip-+ N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan x\right)\right), \color{blue}{\left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \tan x \cdot \tan x\right)}^{2}\right), \left(\color{blue}{1 \cdot 1} - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \tan x \cdot \tan x\right), 2\right), \left(\color{blue}{1 \cdot 1} - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), 2\right), \left(\color{blue}{1} \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   8. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   10. tan-lowering-tan.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), 2\right), \left(1 - \color{blue}{\left(\tan x \cdot \tan x\right)} \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), 2\right), \mathsf{\_.f64}\left(1, \color{blue}{\left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}\right)\right) \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{{\left(1 - {\tan x}^{2}\right)}^{2}}{1 - {\tan x}^{4}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 4\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 56.8%

   \[\leadsto \frac{\color{blue}{1}}{1 - {\tan x}^{4}} \]
8. Add Preprocessing

Alternative 6: 59.1% accurate, 2.0× 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 99.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \color{blue}{\left(1 + \tan x \cdot \tan x\right)}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   5. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 57.8%

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

8. Final simplification 57.8%

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

Alternative 7: 54.8% accurate, 411.0× 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 99.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 53.1%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))