2atan (example 3.5)

Average Error: 14.7 → 14.7

Time: 44.3s

Precision: binary64

Cost: 33472

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1} \left(N + 1\right)\\ t_1 := \frac{\tan^{-1} N}{-4}\\ \frac{t_0}{8} - \left(t_1 + \left(\left(\tan^{-1} N \cdot 1.5 + t_0 \cdot -0.875\right) + t_1\right)\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (atan (+ N 1.0)) (atan N)))

↓

(FPCore (N)
 :precision binary64
 (let* ((t_0 (atan (+ N 1.0))) (t_1 (/ (atan N) -4.0)))
   (- (/ t_0 8.0) (+ t_1 (+ (+ (* (atan N) 1.5) (* t_0 -0.875)) t_1)))))

C

double code(double N) {
	return atan((N + 1.0)) - atan(N);
}

↓

double code(double N) {
	double t_0 = atan((N + 1.0));
	double t_1 = atan(N) / -4.0;
	return (t_0 / 8.0) - (t_1 + (((atan(N) * 1.5) + (t_0 * -0.875)) + t_1));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = atan((n + 1.0d0)) - atan(n)
end function

↓

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    t_0 = atan((n + 1.0d0))
    t_1 = atan(n) / (-4.0d0)
    code = (t_0 / 8.0d0) - (t_1 + (((atan(n) * 1.5d0) + (t_0 * (-0.875d0))) + t_1))
end function

Java

public static double code(double N) {
	return Math.atan((N + 1.0)) - Math.atan(N);
}

↓

public static double code(double N) {
	double t_0 = Math.atan((N + 1.0));
	double t_1 = Math.atan(N) / -4.0;
	return (t_0 / 8.0) - (t_1 + (((Math.atan(N) * 1.5) + (t_0 * -0.875)) + t_1));
}

Python

def code(N):
	return math.atan((N + 1.0)) - math.atan(N)

↓

def code(N):
	t_0 = math.atan((N + 1.0))
	t_1 = math.atan(N) / -4.0
	return (t_0 / 8.0) - (t_1 + (((math.atan(N) * 1.5) + (t_0 * -0.875)) + t_1))

Julia

function code(N)
	return Float64(atan(Float64(N + 1.0)) - atan(N))
end

↓

function code(N)
	t_0 = atan(Float64(N + 1.0))
	t_1 = Float64(atan(N) / -4.0)
	return Float64(Float64(t_0 / 8.0) - Float64(t_1 + Float64(Float64(Float64(atan(N) * 1.5) + Float64(t_0 * -0.875)) + t_1)))
end

MATLAB

function tmp = code(N)
	tmp = atan((N + 1.0)) - atan(N);
end

↓

function tmp = code(N)
	t_0 = atan((N + 1.0));
	t_1 = atan(N) / -4.0;
	tmp = (t_0 / 8.0) - (t_1 + (((atan(N) * 1.5) + (t_0 * -0.875)) + t_1));
end

Wolfram

code[N_] := N[(N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[ArcTan[N], $MachinePrecision]), $MachinePrecision]

↓

code[N_] := Block[{t$95$0 = N[ArcTan[N[(N + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[N], $MachinePrecision] / -4.0), $MachinePrecision]}, N[(N[(t$95$0 / 8.0), $MachinePrecision] - N[(t$95$1 + N[(N[(N[(N[ArcTan[N], $MachinePrecision] * 1.5), $MachinePrecision] + N[(t$95$0 * -0.875), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.7
Target	0.4
Herbie	14.7

\[\tan^{-1} \left(\frac{1}{1 + N \cdot \left(N + 1\right)}\right) \]

Derivation

1. Initial program 14.7

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

2. Applied egg-rr 14.7

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

3. Applied egg-rr 14.8

   \[\leadsto \color{blue}{\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) + \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} N \cdot 2 + \tan^{-1} \left(N + 1\right) \cdot -0.5\right)\right)} \]

4. Simplified 14.7

   \[\leadsto \color{blue}{\frac{\tan^{-1} \left(N + 1\right)}{4} + \left(\frac{\tan^{-1} N}{2} + \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} \left(N + 1\right) \cdot -0.5 + \tan^{-1} N \cdot 2\right)\right)\right)} \]
   Proof

   [Start] 14.8	\[ \left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) + \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} N \cdot 2 + \tan^{-1} \left(N + 1\right) \cdot -0.5\right)\right) \]
   rational_best-simplify-3 [=>] 14.8	\[ \color{blue}{\left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} N \cdot 2 + \tan^{-1} \left(N + 1\right) \cdot -0.5\right)\right) + \left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right)} \]
   rational_best-simplify-3 [=>] 14.8	\[ \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} N \cdot 2 + \tan^{-1} \left(N + 1\right) \cdot -0.5\right)\right) + \color{blue}{\left(\frac{\tan^{-1} N}{2} + \frac{\tan^{-1} \left(N + 1\right)}{4}\right)} \]
   rational_best-simplify-47 [=>] 14.7	\[ \color{blue}{\frac{\tan^{-1} \left(N + 1\right)}{4} + \left(\frac{\tan^{-1} N}{2} + \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \left(\tan^{-1} N \cdot 2 + \tan^{-1} \left(N + 1\right) \cdot -0.5\right)\right)\right)} \]
   rational_best-simplify-3 [=>] 14.7	\[ \frac{\tan^{-1} \left(N + 1\right)}{4} + \left(\frac{\tan^{-1} N}{2} + \left(\left(\frac{\tan^{-1} \left(N + 1\right)}{4} + \frac{\tan^{-1} N}{2}\right) - \color{blue}{\left(\tan^{-1} \left(N + 1\right) \cdot -0.5 + \tan^{-1} N \cdot 2\right)}\right)\right) \]

5. Applied egg-rr 14.7

   \[\leadsto \color{blue}{\frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\frac{\tan^{-1} N}{-2} - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\tan^{-1} N \cdot 1.5 - \left(0 - \tan^{-1} \left(N + 1\right) \cdot -0.75\right)\right)\right)\right)} \]

6. Simplified 14.7

   \[\leadsto \color{blue}{\frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\left(\tan^{-1} \left(N + 1\right) \cdot -0.75 + \tan^{-1} N \cdot 1.5\right) - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \frac{\tan^{-1} N}{-2}\right)\right)} \]
   Proof

   [Start] 14.7	\[ \frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\frac{\tan^{-1} N}{-2} - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\tan^{-1} N \cdot 1.5 - \left(0 - \tan^{-1} \left(N + 1\right) \cdot -0.75\right)\right)\right)\right) \]
   rational_best-simplify-51 [=>] 14.7	\[ \frac{\tan^{-1} \left(N + 1\right)}{8} - \color{blue}{\left(\left(\tan^{-1} N \cdot 1.5 - \left(0 - \tan^{-1} \left(N + 1\right) \cdot -0.75\right)\right) - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \frac{\tan^{-1} N}{-2}\right)\right)} \]
   rational_best-simplify-15 [=>] 14.7	\[ \frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\left(\tan^{-1} N \cdot 1.5 - \color{blue}{\left(-\tan^{-1} \left(N + 1\right) \cdot -0.75\right)}\right) - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \frac{\tan^{-1} N}{-2}\right)\right) \]
   rational_best-simplify-59 [<=] 14.7	\[ \frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\color{blue}{\left(\tan^{-1} \left(N + 1\right) \cdot -0.75 + \tan^{-1} N \cdot 1.5\right)} - \left(\frac{\tan^{-1} \left(N + 1\right)}{8} - \frac{\tan^{-1} N}{-2}\right)\right) \]

7. Applied egg-rr 14.7

   \[\leadsto \frac{\tan^{-1} \left(N + 1\right)}{8} - \color{blue}{\left(\frac{\tan^{-1} N}{-4} + \left(\left(\tan^{-1} N \cdot 1.5 + \tan^{-1} \left(N + 1\right) \cdot -0.875\right) + \frac{\tan^{-1} N}{-4}\right)\right)} \]

8. Final simplification 14.7

   \[\leadsto \frac{\tan^{-1} \left(N + 1\right)}{8} - \left(\frac{\tan^{-1} N}{-4} + \left(\left(\tan^{-1} N \cdot 1.5 + \tan^{-1} \left(N + 1\right) \cdot -0.875\right) + \frac{\tan^{-1} N}{-4}\right)\right) \]

Alternatives

Alternative 1
Error	14.7
Cost	33216

\[\begin{array}{l} t_0 := \tan^{-1} \left(N + 1\right)\\ \left(t_0 + \frac{\tan^{-1} N}{-2}\right) + \left(t_0 - \left(t_0 + \frac{\tan^{-1} N}{2}\right)\right) \end{array} \]

Alternative 2
Error	14.7
Cost	26688

\[\frac{\tan^{-1} N}{8} + \left(\tan^{-1} \left(N + 1\right) - \left(\frac{\tan^{-1} N \cdot 3}{-8} + \tan^{-1} N \cdot 1.5\right)\right) \]

Alternative 3
Error	14.7
Cost	19904

\[\left(\tan^{-1} N \cdot -1.5 + \tan^{-1} \left(N + 1\right)\right) + \tan^{-1} N \cdot 0.5 \]

Alternative 4
Error	14.7
Cost	19904

\[\left(\tan^{-1} \left(N + 1\right) + \tan^{-1} N \cdot -0.875\right) - \tan^{-1} N \cdot 0.125 \]

Alternative 5
Error	14.7
Cost	13120

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

Reproduce

herbie shell --seed 2023099 
(FPCore (N)
  :name "2atan (example 3.5)"
  :precision binary64

  :herbie-target
  (atan (/ 1.0 (+ 1.0 (* N (+ N 1.0)))))

  (- (atan (+ N 1.0)) (atan N)))