Trigonometry B

Average Accuracy: 99.5% → 99.5%

Time: 13.3s

Precision: binary64

Cost: 32512

Math

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

↓

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

FPCore

(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))))

C

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));
}

Julia

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

Wolfram

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]

Derivation

1. Initial program 99.5%

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

2. Applied egg-rr 99.4%

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

   [Start] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   div-sub [=>] 99.4	\[ \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}} \]
   sub-neg [=>] 99.4	\[ \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} + \left(-\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right)} \]
   +-commutative [=>] 99.4	\[ \frac{1}{\color{blue}{\tan x \cdot \tan x + 1}} + \left(-\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right) \]
   fma-def [=>] 99.4	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} + \left(-\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right) \]
   pow2 [=>] 99.4	\[ \frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{\color{blue}{{\tan x}^{2}}}{1 + \tan x \cdot \tan x}\right) \]
   +-commutative [=>] 99.4	\[ \frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{{\tan x}^{2}}{\color{blue}{\tan x \cdot \tan x + 1}}\right) \]
   fma-def [=>] 99.4	\[ \frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{{\tan x}^{2}}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}\right) \]

3. Simplified 99.5%

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

   [Start] 99.4	\[ \frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{{\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   sub-neg [<=] 99.4	\[ \color{blue}{\frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{{\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   div-sub [<=] 99.5	\[ \color{blue}{\frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]

4. Applied egg-rr 99.4%

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

   [Start] 99.5	\[ \frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   frac-2neg [=>] 99.5	\[ \color{blue}{\frac{-\left(1 - {\tan x}^{2}\right)}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   div-inv [=>] 99.4	\[ \color{blue}{\left(-\left(1 - {\tan x}^{2}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   neg-sub0 [=>] 99.4	\[ \color{blue}{\left(0 - \left(1 - {\tan x}^{2}\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   metadata-eval [<=] 99.4	\[ \left(\color{blue}{\log 1} - \left(1 - {\tan x}^{2}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   associate--r- [=>] 99.4	\[ \color{blue}{\left(\left(\log 1 - 1\right) + {\tan x}^{2}\right)} \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   metadata-eval [=>] 99.4	\[ \left(\left(\color{blue}{0} - 1\right) + {\tan x}^{2}\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   metadata-eval [=>] 99.4	\[ \left(\color{blue}{-1} + {\tan x}^{2}\right) \cdot \frac{1}{-\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   neg-sub0 [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\color{blue}{0 - \mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   metadata-eval [<=] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\color{blue}{\log 1} - \mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   fma-udef [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(\tan x \cdot \tan x + 1\right)}} \]
   unpow2 [<=] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\log 1 - \left(\color{blue}{{\tan x}^{2}} + 1\right)} \]
   +-commutative [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\log 1 - \color{blue}{\left(1 + {\tan x}^{2}\right)}} \]
   associate--r+ [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\color{blue}{\left(\log 1 - 1\right) - {\tan x}^{2}}} \]
   metadata-eval [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\left(\color{blue}{0} - 1\right) - {\tan x}^{2}} \]
   metadata-eval [=>] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{\color{blue}{-1} - {\tan x}^{2}} \]

5. Simplified 99.5%

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

   [Start] 99.4	\[ \left(-1 + {\tan x}^{2}\right) \cdot \frac{1}{-1 - {\tan x}^{2}} \]
   *-commutative [=>] 99.4	\[ \color{blue}{\frac{1}{-1 - {\tan x}^{2}} \cdot \left(-1 + {\tan x}^{2}\right)} \]
   associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{1 \cdot \left(-1 + {\tan x}^{2}\right)}{-1 - {\tan x}^{2}}} \]
   +-commutative [=>] 99.5	\[ \frac{1 \cdot \color{blue}{\left({\tan x}^{2} + -1\right)}}{-1 - {\tan x}^{2}} \]
   metadata-eval [<=] 99.5	\[ \frac{1 \cdot \left({\tan x}^{2} + \color{blue}{\left(-1\right)}\right)}{-1 - {\tan x}^{2}} \]
   sub-neg [<=] 99.5	\[ \frac{1 \cdot \color{blue}{\left({\tan x}^{2} - 1\right)}}{-1 - {\tan x}^{2}} \]
   *-lft-identity [=>] 99.5	\[ \frac{\color{blue}{{\tan x}^{2} - 1}}{-1 - {\tan x}^{2}} \]
   unpow2 [=>] 99.5	\[ \frac{\color{blue}{\tan x \cdot \tan x} - 1}{-1 - {\tan x}^{2}} \]
   fma-neg [=>] 99.5	\[ \frac{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}}{-1 - {\tan x}^{2}} \]
   metadata-eval [=>] 99.5	\[ \frac{\mathsf{fma}\left(\tan x, \tan x, \color{blue}{-1}\right)}{-1 - {\tan x}^{2}} \]

6. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	32512

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

Alternative 2
Accuracy	99.5%
Cost	26176

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

Alternative 3
Accuracy	59.7%
Cost	25920

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

Alternative 4
Accuracy	59.7%
Cost	13312

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

Alternative 5
Accuracy	58.9%
Cost	13184

\[\frac{1}{1 - {\tan x}^{4}} \]

Alternative 6
Accuracy	59.7%
Cost	13056

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

Alternative 7
Accuracy	55.5%
Cost	7104

\[1 - \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}} \]

Alternative 8
Accuracy	55.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023126 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))