?

Average Error: 0.3 → 0.4
Time: 11.0s
Precision: binary64
Cost: 26496

?

\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[\frac{1 - \frac{1}{\frac{1}{{\tan x}^{2}}}}{1 + \tan x \cdot \tan x} \]
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (/ 1.0 (/ 1.0 (pow (tan x) 2.0)))) (+ 1.0 (* (tan x) (tan x)))))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return (1.0 - (1.0 / (1.0 / pow(tan(x), 2.0)))) / (1.0 + (tan(x) * tan(x)));
}

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
    code = (1.0d0 - (1.0d0 / (1.0d0 / (tan(x) ** 2.0d0)))) / (1.0d0 + (tan(x) * tan(x)))
end function

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) {
	return (1.0 - (1.0 / (1.0 / Math.pow(Math.tan(x), 2.0)))) / (1.0 + (Math.tan(x) * Math.tan(x)));
}

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

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)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(1.0 / (tan(x) ^ 2.0)))) / Float64(1.0 + Float64(tan(x) * tan(x))))
end

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

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_] := N[(N[(1.0 - N[(1.0 / N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{1}{\frac{1}{{\tan x}^{2}}}}{1 + \tan x \cdot \tan x}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.3

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

  2. Applied egg-rr0.4

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

  3. Final simplification0.4

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

Alternatives

Alternative 1
Error0.3
Cost26176
\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{t_0 + -1}{-1 - t_0} \end{array} \]
Alternative 2
Error26.0
Cost13312
\[\frac{-1}{\frac{1}{{\tan x}^{2} + -1}} \]
Alternative 3
Error26.5
Cost13184
\[\frac{1}{1 - {\tan x}^{4}} \]
Alternative 4
Error26.0
Cost13120
\[1 - \tan x \cdot \tan x \]
Alternative 5
Error28.7
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 2023057 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))