Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 6.7

Time: 25.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ t_2 := \frac{\frac{\frac{\ell}{k} \cdot \left(\ell \cdot t_1\right)}{k} \cdot 2}{{\sin k}^{2}}\\ \mathbf{if}\;k \leq -2.2 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \frac{t_1 \cdot 2}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (cos k) t))
        (t_2 (/ (* (/ (* (/ l k) (* l t_1)) k) 2.0) (pow (sin k) 2.0))))
   (if (<= k -2.2e-60)
     t_2
     (if (<= k 1.3e-98)
       (* (* (/ l k) (/ (/ l k) (sin k))) (/ (* t_1 2.0) (sin k)))
       t_2))))

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 = cos(k) / t;
	double t_2 = ((((l / k) * (l * t_1)) / k) * 2.0) / pow(sin(k), 2.0);
	double tmp;
	if (k <= -2.2e-60) {
		tmp = t_2;
	} else if (k <= 1.3e-98) {
		tmp = ((l / k) * ((l / k) / sin(k))) * ((t_1 * 2.0) / sin(k));
	} else {
		tmp = t_2;
	}
	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 = cos(k) / t
    t_2 = ((((l / k) * (l * t_1)) / k) * 2.0d0) / (sin(k) ** 2.0d0)
    if (k <= (-2.2d-60)) then
        tmp = t_2
    else if (k <= 1.3d-98) then
        tmp = ((l / k) * ((l / k) / sin(k))) * ((t_1 * 2.0d0) / sin(k))
    else
        tmp = t_2
    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 = Math.cos(k) / t;
	double t_2 = ((((l / k) * (l * t_1)) / k) * 2.0) / Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= -2.2e-60) {
		tmp = t_2;
	} else if (k <= 1.3e-98) {
		tmp = ((l / k) * ((l / k) / Math.sin(k))) * ((t_1 * 2.0) / Math.sin(k));
	} else {
		tmp = t_2;
	}
	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 = math.cos(k) / t
	t_2 = ((((l / k) * (l * t_1)) / k) * 2.0) / math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= -2.2e-60:
		tmp = t_2
	elif k <= 1.3e-98:
		tmp = ((l / k) * ((l / k) / math.sin(k))) * ((t_1 * 2.0) / math.sin(k))
	else:
		tmp = t_2
	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(cos(k) / t)
	t_2 = Float64(Float64(Float64(Float64(Float64(l / k) * Float64(l * t_1)) / k) * 2.0) / (sin(k) ^ 2.0))
	tmp = 0.0
	if (k <= -2.2e-60)
		tmp = t_2;
	elseif (k <= 1.3e-98)
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / sin(k))) * Float64(Float64(t_1 * 2.0) / sin(k)));
	else
		tmp = t_2;
	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 = cos(k) / t;
	t_2 = ((((l / k) * (l * t_1)) / k) * 2.0) / (sin(k) ^ 2.0);
	tmp = 0.0;
	if (k <= -2.2e-60)
		tmp = t_2;
	elseif (k <= 1.3e-98)
		tmp = ((l / k) * ((l / k) / sin(k))) * ((t_1 * 2.0) / sin(k));
	else
		tmp = t_2;
	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[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.2e-60], t$95$2, If[LessEqual[k, 1.3e-98], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < -2.1999999999999999e-60 or 1.30000000000000007e-98 < k

   1. Initial program 46.1

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

   2. Simplified 37.9

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

   3. Taylor expanded in t around 0 19.9

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

   4. Simplified 17.3

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

   5. Taylor expanded in k around inf 19.9

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

   6. Simplified 15.6

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

   7. Applied egg-rr 6.5

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

   if -2.1999999999999999e-60 < k < 1.30000000000000007e-98

   1. Initial program 63.8

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

   2. Simplified 63.2

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

   3. Taylor expanded in t around 0 53.4

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

   4. Simplified 53.9

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

   5. Taylor expanded in k around inf 53.4

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

   6. Simplified 33.0

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

   7. Applied egg-rr 11.9

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

   8. Applied egg-rr 8.6

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{t}\right)}{k} \cdot 2}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right) \cdot \frac{\frac{\cos k}{t} \cdot 2}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \left(\ell \cdot \frac{\cos k}{t}\right)}{k} \cdot 2}{{\sin k}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(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))))