Trigonometry B

Average Error: 0.3 → 0.3

Time: 10.7s

Precision: binary64

Cost: 26176

Math

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

↓

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

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))) (/ (+ t_0 -1.0) (- -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 (t_0 + -1.0) / (-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 = (t_0 + (-1.0d0)) / ((-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 (t_0 + -1.0) / (-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 (t_0 + -1.0) / (-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(t_0 + -1.0) / 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 = (t_0 + -1.0) / (-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[(t$95$0 + -1.0), $MachinePrecision] / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   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))))): 3 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.4

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

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\tan x}^{2} + -1}{-1 - {\tan x}^{2}}} \]
   Proof
   (/.f64 (+.f64 (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))): 0 points increase in error, 1 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)))))): 4 points increase in error, 0 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, 4 points decrease in error

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	25.9
Cost	13056

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

Alternative 2
Error	28.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))