Average Error: 0.3 → 0.3
Time: 11.1s
Precision: binary64
Cost: 32512
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}} \]
(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 (pow (tan x) 2.0))))
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 - pow(tan(x), 2.0));
}

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) / Float64(-1.0 - (tan(x) ^ 2.0)))
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[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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}^{2}}

Error

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
    Proof
    (/.f64 (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 x))) (fma.f64 (tan.f64 x) (tan.f64 x) 1)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 x))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (tan.f64 x) (tan.f64 x)) 1))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 x))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (tan.f64 x) (tan.f64 x))))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.4

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)} - 1} \]

  4. Simplified0.3

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}}} \]
    Proof
    (/.f64 (fma.f64 (tan.f64 x) (tan.f64 x) -1) (-.f64 -1 (pow.f64 (tan.f64 x) 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (tan.f64 x) (tan.f64 x)) -1)) (-.f64 -1 (pow.f64 (tan.f64 x) 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (tan.f64 x) 2)) -1) (-.f64 -1 (pow.f64 (tan.f64 x) 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 -1 (pow.f64 (tan.f64 x) 2))) (-.f64 -1 (pow.f64 (tan.f64 x) 2))): 5 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (+.f64 -1 (pow.f64 (tan.f64 x) 2)) (-.f64 -1 (pow.f64 (tan.f64 x) 2)))))): 5 points increase in error, 1 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (+.f64 -1 (pow.f64 (tan.f64 x) 2)) (-.f64 -1 (pow.f64 (tan.f64 x) 2))))) 1)): 0 points increase in error, 6 points decrease in error

  5. Final simplification0.3

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

Alternatives

Alternative 1
Error26.4
Cost26249
\[\begin{array}{l} t_0 := -1 - {\tan x}^{2}\\ \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t_0}\\ \end{array} \]
Alternative 2
Error0.3
Cost26176
\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{-1 + t_0}{-1 - t_0} \end{array} \]
Alternative 3
Error29.0
Cost13184
\[\frac{-1}{-1 - {\tan x}^{2}} \]
Alternative 4
Error26.4
Cost13056
\[1 - {\tan x}^{2} \]
Alternative 5
Error29.2
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022340 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))