Trigonometry B

Average Error: 0.3 → 0.3

Time: 12.6s

Precision: binary64

Cost: 26176

Math

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

↓

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

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))) (/ (- 1.0 t_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 (1.0 - t_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 = (1.0d0 - t_0) / (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 (1.0 - t_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 (1.0 - t_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(1.0 - t_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 = (1.0 - t_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[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.4

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

3. Simplified 0.3

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

   [Start] 0.4	\[ \frac{1}{1 + \tan x \cdot \tan x} + \left(-\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right) \]
   sub-neg [<=] 0.4	\[ \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]
   div-sub [<=] 0.3	\[ \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]
   /-rgt-identity [<=] 0.3	\[ \frac{\color{blue}{\frac{1 - \tan x \cdot \tan x}{1}}}{1 + \tan x \cdot \tan x} \]
   associate-/r* [<=] 0.3	\[ \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 \cdot \left(1 + \tan x \cdot \tan x\right)}} \]
   unpow2 [<=] 0.3	\[ \frac{1 - \color{blue}{{\tan x}^{2}}}{1 \cdot \left(1 + \tan x \cdot \tan x\right)} \]
   *-lft-identity [=>] 0.3	\[ \frac{1 - {\tan x}^{2}}{\color{blue}{1 + \tan x \cdot \tan x}} \]
   unpow2 [<=] 0.3	\[ \frac{1 - {\tan x}^{2}}{1 + \color{blue}{{\tan x}^{2}}} \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	28.8
Cost	13184

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

Alternative 2
Error	26.8
Cost	13184

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

Alternative 3
Error	29.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))