Average Error: 0.3 → 0.4
Time: 5.6s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[{\left(\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}\right)}^{-1} \]
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (pow (/ (fma (tan x) (tan x) 1.0) (fma (tan x) (- (tan x)) 1.0)) -1.0))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return pow((fma(tan(x), tan(x), 1.0) / fma(tan(x), -tan(x), 1.0)), -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(fma(tan(x), tan(x), 1.0) / fma(tan(x), Float64(-tan(x)), 1.0)) ^ -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[Power[N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022192 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))