Average Error: 0.3 → 0.3
Time: 8.1s
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)}

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))): 5 points increase in error, 3 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

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

  4. Simplified0.3

    \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
    Proof
    (/.f64 (-.f64 1 (pow.f64 (tan.f64 x) 2)) (fma.f64 (tan.f64 x) (tan.f64 x) 1)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (pow.f64 (tan.f64 x) 2)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (tan.f64 x) (tan.f64 x)) 1))): 5 points increase in error, 3 points decrease in error
    (/.f64 (-.f64 1 (pow.f64 (tan.f64 x) 2)) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (tan.f64 x) 2)) 1)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 1 (pow.f64 (tan.f64 x) 2)) (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (tan.f64 x) 2)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 (+.f64 1 (pow.f64 (tan.f64 x) 2))) (/.f64 (pow.f64 (tan.f64 x) 2) (+.f64 1 (pow.f64 (tan.f64 x) 2))))): 36 points increase in error, 24 points decrease in error
    (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 1 (+.f64 1 (pow.f64 (tan.f64 x) 2))) (neg.f64 (/.f64 (pow.f64 (tan.f64 x) 2) (+.f64 1 (pow.f64 (tan.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost26176
\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t_0}{1 + t_0} \end{array} \]
Alternative 2
Error26.4
Cost13056
\[1 - {\tan x}^{2} \]
Alternative 3
Error29.2
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))