Trigonometry B

Average Accuracy: 99.5% → 99.5%

Time: 11.3s

Precision: binary64

Cost: 26176

Math

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

↓

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}

↓

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) * tan(x))) / (1.0d0 + (tan(x) * tan(x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	return (1.0 - (Math.tan(x) * Math.tan(x))) / (1.0 + (Math.tan(x) * Math.tan(x)));
}

↓

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	return (1.0 - (math.tan(x) * math.tan(x))) / (1.0 + (math.tan(x) * math.tan(x)))

↓

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_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)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
end

↓

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_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_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

2. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   +-commutative [=>] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\tan x \cdot \tan x + 1}} \]
   fma-def [=>] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]

3. Applied egg-rr 99.2%

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

   [Start] 99.5	\[ \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
   div-sub [=>] 99.4	\[ \color{blue}{\frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   sub-neg [=>] 99.4	\[ \color{blue}{\frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)} \]
   add-sqr-sqrt [=>] 99.2	\[ \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   pow2 [=>] 99.2	\[ \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)}^{2}}} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   pow-flip [=>] 99.2	\[ \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)}^{\left(-2\right)}} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   fma-udef [=>] 99.2	\[ {\left(\sqrt{\color{blue}{\tan x \cdot \tan x + 1}}\right)}^{\left(-2\right)} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   +-commutative [<=] 99.2	\[ {\left(\sqrt{\color{blue}{1 + \tan x \cdot \tan x}}\right)}^{\left(-2\right)} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   hypot-1-def [=>] 99.2	\[ {\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{\left(-2\right)} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   metadata-eval [=>] 99.2	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{\color{blue}{-2}} + \left(-\frac{\tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   add-sqr-sqrt [=>] 99.1	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2} + \left(-\frac{\tan x \cdot \tan x}{\color{blue}{\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}}\right) \]
   times-frac [=>] 99.1	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2} + \left(-\color{blue}{\frac{\tan x}{\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \cdot \frac{\tan x}{\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}}\right) \]
   pow2 [=>] 99.1	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2} + \left(-\color{blue}{{\left(\frac{\tan x}{\sqrt{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}\right)}^{2}}\right) \]

4. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}} \]
   Proof

   [Start] 99.2	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2} + \left(-{\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\right) \]
   sub-neg [<=] 99.2	\[ \color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2} - {\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}} \]
   metadata-eval [<=] 99.2	\[ {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{\color{blue}{\left(2 \cdot -1\right)}} - {\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2} \]
   pow-sqr [<=] 99.1	\[ \color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-1}} - {\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2} \]
   unpow-1 [=>] 99.1	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}} \cdot {\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-1} - {\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2} \]
   unpow-1 [=>] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}} - {\left(\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2} \]
   unpow2 [=>] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \color{blue}{\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)}} \]
   associate-*r/ [=>] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \color{blue}{\frac{\frac{\tan x}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \tan x}{\mathsf{hypot}\left(1, \tan x\right)}} \]
   associate-*l/ [=>] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \frac{\color{blue}{\frac{\tan x \cdot \tan x}{\mathsf{hypot}\left(1, \tan x\right)}}}{\mathsf{hypot}\left(1, \tan x\right)} \]
   unpow2 [<=] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \frac{\frac{\color{blue}{{\tan x}^{2}}}{\mathsf{hypot}\left(1, \tan x\right)}}{\mathsf{hypot}\left(1, \tan x\right)} \]
   *-rgt-identity [<=] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \frac{\color{blue}{\frac{{\tan x}^{2}}{\mathsf{hypot}\left(1, \tan x\right)} \cdot 1}}{\mathsf{hypot}\left(1, \tan x\right)} \]
   associate-*r/ [<=] 99.1	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \color{blue}{\frac{{\tan x}^{2}}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \tan x\right)}} \]
   distribute-rgt-out-- [=>] 99.2	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)} - \frac{{\tan x}^{2}}{\mathsf{hypot}\left(1, \tan x\right)}\right)} \]
   div-sub [<=] 99.2	\[ \frac{1}{\mathsf{hypot}\left(1, \tan x\right)} \cdot \color{blue}{\frac{1 - {\tan x}^{2}}{\mathsf{hypot}\left(1, \tan x\right)}} \]

5. Final simplification 99.5%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]

Alternatives

Alternative 1
Accuracy	54.8%
Cost	13184

\[\frac{1}{1 + {\tan x}^{2}} \]

Alternative 2
Accuracy	57.9%
Cost	13184

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

Alternative 3
Accuracy	58.7%
Cost	13056

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

Alternative 4
Accuracy	54.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023142 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))