Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

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 99.6%

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

   1. div-sub 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. add-sqr-sqrt 99.3%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.2%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (/ -1.0 (+ (pow (tan x) 2.0) 1.0))
   (pow (/ 1.0 (hypot 1.0 (tan x))) 2.0)))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 / (pow(tan(x), 2.0) + 1.0);
	} else {
		tmp = pow((1.0 / hypot(1.0, tan(x))), 2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 / (Math.pow(Math.tan(x), 2.0) + 1.0);
	} else {
		tmp = Math.pow((1.0 / Math.hypot(1.0, Math.tan(x))), 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 / (math.pow(math.tan(x), 2.0) + 1.0)
	else:
		tmp = math.pow((1.0 / math.hypot(1.0, math.tan(x))), 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 / Float64((tan(x) ^ 2.0) + 1.0));
	else
		tmp = Float64(1.0 / hypot(1.0, tan(x))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 / ((tan(x) ^ 2.0) + 1.0);
	else
		tmp = (1.0 / hypot(1.0, tan(x))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   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. div-sub 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. add-sqr-sqrt 98.6%

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

      4. prod-diff 98.7%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 17.0%

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

   if -1 < (tan.f64 x) < 1

   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. div-sub 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. prod-diff 99.6%

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

   4. Applied egg-rr 99.4%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 74.4%

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

      1. add-sqr-sqrt 74.4%

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

      2. pow2 74.4%

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

   9. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2} + 1\\ \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;\frac{-1}{\sqrt[3]{{t\_0}^{3}}}\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (pow (tan x) 2.0) 1.0)))
   (if (<= (tan x) -1.0)
     (/ -1.0 (cbrt (pow t_0 3.0)))
     (if (<= (tan x) 1.0)
       (pow (/ 1.0 (hypot 1.0 (tan x))) 2.0)
       (/ -1.0 t_0)))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0) + 1.0;
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -1.0 / cbrt(pow(t_0, 3.0));
	} else if (tan(x) <= 1.0) {
		tmp = pow((1.0 / hypot(1.0, tan(x))), 2.0);
	} 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) + 1.0;
	double tmp;
	if (Math.tan(x) <= -1.0) {
		tmp = -1.0 / Math.cbrt(Math.pow(t_0, 3.0));
	} else if (Math.tan(x) <= 1.0) {
		tmp = Math.pow((1.0 / Math.hypot(1.0, Math.tan(x))), 2.0);
	} else {
		tmp = -1.0 / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64((tan(x) ^ 2.0) + 1.0)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = Float64(-1.0 / cbrt((t_0 ^ 3.0)));
	elseif (tan(x) <= 1.0)
		tmp = Float64(1.0 / hypot(1.0, tan(x))) ^ 2.0;
	else
		tmp = Float64(-1.0 / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[(-1.0 / N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[x], $MachinePrecision], 1.0], N[Power[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(-1.0 / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 x) < -1

   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. div-sub 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. add-sqr-sqrt 98.8%

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

      4. prod-diff 98.9%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 17.2%

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

   if -1 < (tan.f64 x) < 1

   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. div-sub 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. prod-diff 99.6%

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

   4. Applied egg-rr 99.4%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 74.4%

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

      1. add-sqr-sqrt 74.4%

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

      2. pow2 74.4%

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

   9. Applied egg-rr 74.4%

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

   if 1 < (tan.f64 x)

   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. div-sub 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. add-sqr-sqrt 98.5%

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

      4. prod-diff 98.4%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 16.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;\frac{-1}{\sqrt[3]{{\left({\tan x}^{2} + 1\right)}^{3}}}\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (/ -1.0 (+ (pow (tan x) 2.0) 1.0))
   (pow (hypot 1.0 (tan x)) -2.0)))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 / (pow(tan(x), 2.0) + 1.0);
	} else {
		tmp = pow(hypot(1.0, tan(x)), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 / (Math.pow(Math.tan(x), 2.0) + 1.0);
	} else {
		tmp = Math.pow(Math.hypot(1.0, Math.tan(x)), -2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 / (math.pow(math.tan(x), 2.0) + 1.0)
	else:
		tmp = math.pow(math.hypot(1.0, math.tan(x)), -2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 / Float64((tan(x) ^ 2.0) + 1.0));
	else
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 / ((tan(x) ^ 2.0) + 1.0);
	else
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   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. div-sub 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. add-sqr-sqrt 98.6%

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

      4. prod-diff 98.7%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 17.0%

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

   if -1 < (tan.f64 x) < 1

   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. div-sub 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. prod-diff 99.6%

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

   4. Applied egg-rr 99.4%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 74.4%

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

      1. frac-2neg 74.4%

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

      2. metadata-eval 74.4%

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

      3. add-sqr-sqrt 74.4%

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

   9. Applied egg-rr 74.4%

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

       1. unpow-1 74.4%

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

       2. unpow-1 74.4%

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

       3. pow-sqr 74.4%

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

       4. metadata-eval 74.4%

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

   11. Simplified 74.4%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-1}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (/ -1.0 (+ (pow (tan x) 2.0) 1.0))
   1.0))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 / (pow(tan(x), 2.0) + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((tan(x) <= (-1.0d0)) .or. (.not. (tan(x) <= 1.0d0))) then
        tmp = (-1.0d0) / ((tan(x) ** 2.0d0) + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 / (math.pow(math.tan(x), 2.0) + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 / Float64((tan(x) ^ 2.0) + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 / ((tan(x) ^ 2.0) + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   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. div-sub 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. add-sqr-sqrt 98.6%

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

      4. prod-diff 98.7%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 17.0%

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

   if -1 < (tan.f64 x) < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

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

Alternative 6: 59.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
     (/ -1.0 (+ t_0 1.0))
     (/ -1.0 (- -1.0 t_0)))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 / (t_0 + 1.0);
	} else {
		tmp = -1.0 / (-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) <= (-1.0d0)) .or. (.not. (tan(x) <= 1.0d0))) then
        tmp = (-1.0d0) / (t_0 + 1.0d0)
    else
        tmp = (-1.0d0) / ((-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) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 / (t_0 + 1.0);
	} else {
		tmp = -1.0 / (-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) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 / (t_0 + 1.0)
	else:
		tmp = -1.0 / (-1.0 - t_0)
	return tmp

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 / Float64(t_0 + 1.0));
	else
		tmp = Float64(-1.0 / 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) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 / (t_0 + 1.0);
	else
		tmp = -1.0 / (-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[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   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. div-sub 98.9%

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

      2. *-un-lft-identity 98.9%

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

      3. add-sqr-sqrt 98.6%

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

      4. prod-diff 98.7%

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

   4. Applied egg-rr 98.6%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 1.6%

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

   9. Applied egg-rr 17.0%

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

   if -1 < (tan.f64 x) < 1

   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. div-sub 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. prod-diff 99.6%

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

   4. Applied egg-rr 99.4%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 74.4%

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

4. Final simplification 59.8%

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

Alternative 7: 55.1% 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 99.6%

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

3. Taylor expanded in x around 0 55.5%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 55.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))