Average Error: 0.3 → 0.3
Time: 8.7s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[\frac{\mathsf{fma}\left(-\tan x, \tan x, 1\right)}{1 + \tan x \cdot \tan x} \]
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (fma (- (tan x)) (tan x) 1.0) (+ 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 fma(-tan(x), tan(x), 1.0) / (1.0 + (tan(x) * 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(fma(Float64(-tan(x)), tan(x), 1.0) / Float64(1.0 + Float64(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[((-N[Tan[x], $MachinePrecision]) * N[Tan[x], $MachinePrecision] + 1.0), $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{\mathsf{fma}\left(-\tan x, \tan x, 1\right)}{1 + \tan x \cdot \tan x}

Error

Bits error versus x

Derivation

  1. Initial program 0.3

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

  2. Applied egg-rr0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))