Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 10.6

Time: 16.7s

Precision: binary64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{t} \cdot \cos k\\ \mathbf{if}\;t \leq -2.1469583486825252 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot t_1}{t_2}}\\ \mathbf{elif}\;t \leq 3.1692035794907957 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t_2}{t_1 \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (* (/ l t) (cos k))))
   (if (<= t -2.1469583486825252e-116)
     (/ 2.0 (/ (* (* (sin k) (* t (/ (* t (sin k)) l))) t_1) t_2))
     (if (<= t 3.1692035794907957e-171)
       (/
        2.0
        (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (cos k) (pow l 2.0))))
       (/
        2.0
        (/ 1.0 (/ t_2 (* t_1 (* (sin k) (* t (* (sin k) (/ t l))))))))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = (l / t) * cos(k);
	double tmp;
	if (t <= -2.1469583486825252e-116) {
		tmp = 2.0 / (((sin(k) * (t * ((t * sin(k)) / l))) * t_1) / t_2);
	} else if (t <= 3.1692035794907957e-171) {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / (1.0 / (t_2 / (t_1 * (sin(k) * (t * (sin(k) * (t / l)))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 + ((k / t) ** 2.0d0)
    t_2 = (l / t) * cos(k)
    if (t <= (-2.1469583486825252d-116)) then
        tmp = 2.0d0 / (((sin(k) * (t * ((t * sin(k)) / l))) * t_1) / t_2)
    else if (t <= 3.1692035794907957d-171) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (t * (sin(k) ** 2.0d0))) / (cos(k) * (l ** 2.0d0)))
    else
        tmp = 2.0d0 / (1.0d0 / (t_2 / (t_1 * (sin(k) * (t * (sin(k) * (t / l)))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = (l / t) * Math.cos(k);
	double tmp;
	if (t <= -2.1469583486825252e-116) {
		tmp = 2.0 / (((Math.sin(k) * (t * ((t * Math.sin(k)) / l))) * t_1) / t_2);
	} else if (t <= 3.1692035794907957e-171) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (1.0 / (t_2 / (t_1 * (Math.sin(k) * (t * (Math.sin(k) * (t / l)))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

↓

def code(t, l, k):
	t_1 = 2.0 + math.pow((k / t), 2.0)
	t_2 = (l / t) * math.cos(k)
	tmp = 0
	if t <= -2.1469583486825252e-116:
		tmp = 2.0 / (((math.sin(k) * (t * ((t * math.sin(k)) / l))) * t_1) / t_2)
	elif t <= 3.1692035794907957e-171:
		tmp = 2.0 / ((math.pow(k, 2.0) * (t * math.pow(math.sin(k), 2.0))) / (math.cos(k) * math.pow(l, 2.0)))
	else:
		tmp = 2.0 / (1.0 / (t_2 / (t_1 * (math.sin(k) * (t * (math.sin(k) * (t / l)))))))
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(Float64(l / t) * cos(k))
	tmp = 0.0
	if (t <= -2.1469583486825252e-116)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * Float64(t * Float64(Float64(t * sin(k)) / l))) * t_1) / t_2));
	elseif (t <= 3.1692035794907957e-171)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(1.0 / Float64(t_2 / Float64(t_1 * Float64(sin(k) * Float64(t * Float64(sin(k) * Float64(t / l))))))));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

↓

function tmp_2 = code(t, l, k)
	t_1 = 2.0 + ((k / t) ^ 2.0);
	t_2 = (l / t) * cos(k);
	tmp = 0.0;
	if (t <= -2.1469583486825252e-116)
		tmp = 2.0 / (((sin(k) * (t * ((t * sin(k)) / l))) * t_1) / t_2);
	elseif (t <= 3.1692035794907957e-171)
		tmp = 2.0 / (((k ^ 2.0) * (t * (sin(k) ^ 2.0))) / (cos(k) * (l ^ 2.0)));
	else
		tmp = 2.0 / (1.0 / (t_2 / (t_1 * (sin(k) * (t * (sin(k) * (t / l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / t), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.1469583486825252e-116], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(t * N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1692035794907957e-171], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(1.0 / N[(t$95$2 / N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.14695834868252524e-116

   1. Initial program 24.4

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Simplified 24.4

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. Applied unpow3_binary64 24.4

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   4. Applied times-frac_binary64 17.1

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied associate-*l*_binary64 14.4

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   6. Applied associate-/l*_binary64 9.8

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   7. Applied tan-quot_binary64 9.8

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   8. Applied associate-*l/_binary64 8.7

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied frac-times_binary64 7.0

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   10. Applied associate-*l/_binary64 6.1

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t} \cdot \cos k}}} \]

   11. Taylor expanded in t around 0 6.0

       \[\leadsto \frac{2}{\frac{\left(\left(t \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t} \cdot \cos k}} \]

   if -2.14695834868252524e-116 < t < 3.16920357949079572e-171

   1. Initial program 64.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. Taylor expanded in t around 0 27.6

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}} \]

   if 3.16920357949079572e-171 < t

   1. Initial program 27.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Simplified 27.2

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. Applied unpow3_binary64 27.2

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   4. Applied times-frac_binary64 18.1

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied associate-*l*_binary64 16.1

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   6. Applied associate-/l*_binary64 11.4

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   7. Applied tan-quot_binary64 11.5

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   8. Applied associate-*l/_binary64 10.4

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   9. Applied frac-times_binary64 9.0

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   10. Applied associate-*l/_binary64 8.0

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t} \cdot \cos k}}} \]

   11. Applied clear-num_binary64 8.0

       \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t} \cdot \cos k}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1469583486825252 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 3.1692035794907957 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t} \cdot \cos k}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))