Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.2s

Alternatives: 5

Speedup: N/A×

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. clear-num 99.4%

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

   2. inv-pow 99.4%

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

   3. add-sqr-sqrt 99.2%

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

   4. pow2 99.2%

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

   5. hypot-1-def 99.2%

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

   6. pow2 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. hypot-1-def 99.2%

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

   3. pow2 99.2%

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

   4. sqrt-pow2 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow1 99.4%

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

   7. clear-num 99.6%

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

   8. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 57.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
   (/ 1.0 (/ (fma x x 1.0) (- 1.0 (pow x 2.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (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. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. add-sqr-sqrt 99.4%

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

      4. pow2 99.4%

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

      5. hypot-1-def 99.4%

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

      6. pow2 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

      2. hypot-1-def 99.4%

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

      3. pow2 99.4%

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

      4. sqrt-pow2 99.6%

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

      5. metadata-eval 99.6%

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

      6. pow1 99.6%

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

      7. clear-num 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 72.9%

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

   if 1 < (*.f64 (tan.f64 x) (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. clear-num 99.0%

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

      2. inv-pow 99.0%

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

      3. add-sqr-sqrt 98.7%

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

      4. pow2 98.7%

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

      5. hypot-1-def 98.6%

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

      6. pow2 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. unpow-1 98.6%

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

      2. unpow2 98.6%

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

      3. hypot-1-def 98.7%

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

      4. pow2 98.7%

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

      5. hypot-1-def 98.7%

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

      6. pow2 98.7%

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

      7. add-sqr-sqrt 99.0%

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

      8. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0 4.5%

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

   8. Taylor expanded in x around 0 11.2%

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

      1. +-commutative 11.2%

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

      2. unpow2 11.2%

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

      3. fma-def 11.2%

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

   10. Simplified 11.2%

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

4. Final simplification 56.3%

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

Alternative 3: 55.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $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. clear-num 99.4%

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

   2. inv-pow 99.4%

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

   3. add-sqr-sqrt 99.2%

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

   4. pow2 99.2%

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

   5. hypot-1-def 99.2%

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

   6. pow2 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. hypot-1-def 99.2%

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

   3. pow2 99.2%

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

   4. sqrt-pow2 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow1 99.4%

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

   7. clear-num 99.6%

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

   8. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 53.7%

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

8. Final simplification 53.7%

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

Alternative 4: 58.5% 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 99.6%

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

   1. flip-+ 99.5%

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

   2. associate-/r/ 99.4%

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

   3. pow2 99.4%

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

   4. metadata-eval 99.4%

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

   5. pow2 99.4%

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

   6. pow2 99.4%

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

   7. pow-prod-up 99.3%

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

   8. metadata-eval 99.3%

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

   9. pow2 99.3%

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

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 - {\tan x}^{4}} \cdot \left(1 - {\tan x}^{2}\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 99.3%

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

   2. unpow2 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in x around 0 57.0%

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

8. Final simplification 57.0%

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

Alternative 5: 55.0% 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. Step-by-step derivation

   1. clear-num 99.4%

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

   2. inv-pow 99.4%

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

   3. add-sqr-sqrt 99.2%

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

   4. pow2 99.2%

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

   5. hypot-1-def 99.2%

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

   6. pow2 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow-1 99.2%

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

   2. hypot-1-def 99.2%

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

   3. pow2 99.2%

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

   4. sqrt-pow2 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow1 99.4%

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

   7. clear-num 99.6%

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

   8. +-commutative 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0 53.3%

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

8. Final simplification 53.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))