Trigonometry B

Average Accuracy: 99.5% → 99.5%

Time: 12.9s

Precision: binary64

Cost: 26176

Math

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

↓

\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{t_0 + -1}{-1 - t_0} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (+ t_0 -1.0) (- -1.0 t_0))))

C

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

↓

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (t_0 + -1.0) / (-1.0 - t_0);
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (t_0 + (-1.0d0)) / ((-1.0d0) - t_0)
end function

Java

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

↓

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (t_0 + -1.0) / (-1.0 - t_0);
}

Python

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

↓

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (t_0 + -1.0) / (-1.0 - t_0)

Julia

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

↓

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(t_0 + -1.0) / Float64(-1.0 - t_0))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (t_0 + -1.0) / (-1.0 - t_0);
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

2. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   +-commutative [=>] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\tan x \cdot \tan x + 1}} \]
   fma-def [=>] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]

3. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)} - 1} \]
   Proof

   [Start] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   expm1-log1p-u [=>] 99.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)\right)} \]
   expm1-udef [=>] 99.3	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)} - 1} \]

4. Simplified 99.5%

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

   [Start] 99.3	\[ e^{\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)} - 1 \]
   expm1-def [=>] 99.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)\right)} \]
   expm1-log1p [=>] 99.5	\[ \color{blue}{\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}} \]
   +-commutative [=>] 99.5	\[ \frac{\color{blue}{{\tan x}^{2} + -1}}{-1 - {\tan x}^{2}} \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	59.0%
Cost	26249

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

Alternative 2
Accuracy	59.5%
Cost	14208

\[\frac{{\tan x}^{2} + -1}{-1 + \frac{-1}{\left(\frac{1}{x \cdot x} + \left(x \cdot x\right) \cdot 0.06666666666666667\right) + -0.6666666666666666}} \]

Alternative 3
Accuracy	55.2%
Cost	13184

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

Alternative 4
Accuracy	58.2%
Cost	13056

\[1 - {\tan x}^{4} \]

Alternative 5
Accuracy	59.0%
Cost	13056

\[1 - {\tan x}^{2} \]

Alternative 6
Accuracy	54.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))