Average Error: 0.3 → 0.3

Time: 6.3s

Precision: binary64

Cost: 32512

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

↓

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

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

↓

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

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

↓

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

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 - (tan(x) ^ 2.0)) / fma(tan(x), tan(x), 1.0))
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[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

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

↓

\frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}

Derivation

Initial program 0.3
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
Applied egg-rr0.5
\[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}} \]
Applied egg-rr0.5
\[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} \]
Applied egg-rr0.3
\[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
Final simplification0.3
\[\leadsto \frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	26176

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

Alternative 2
Error	0.3
Cost	26176

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

Alternative 3
Error	26.2
Cost	25920

\[\sqrt[3]{{\left(1 - {\tan x}^{2}\right)}^{3}} \]

Alternative 4
Error	26.2
Cost	13056

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

Alternative 5
Error	28.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))