Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 2.3

Time: 26.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -1.1598924007845897 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{{\sin k}^{2} \cdot \frac{t}{\cos k}}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 6.310209504498685 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}}{\frac{k}{\ell}}\\ \end{array} \]

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
 (if (<= k -1.1598924007845897e-155)
   (* 2.0 (/ (/ (/ l k) (* (pow (sin k) 2.0) (/ t (cos k)))) (/ k l)))
   (if (<= k 6.310209504498685e-147)
     (* 2.0 (/ (/ (* (cos k) (/ (pow (/ l k) 2.0) t)) (sin k)) (sin k)))
     (* 2.0 (/ (/ (/ l k) (/ t (* (cos k) (pow (sin k) -2.0)))) (/ k 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 tmp;
	if (k <= -1.1598924007845897e-155) {
		tmp = 2.0 * (((l / k) / (pow(sin(k), 2.0) * (t / cos(k)))) / (k / l));
	} else if (k <= 6.310209504498685e-147) {
		tmp = 2.0 * (((cos(k) * (pow((l / k), 2.0) / t)) / sin(k)) / sin(k));
	} else {
		tmp = 2.0 * (((l / k) / (t / (cos(k) * pow(sin(k), -2.0)))) / (k / 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) :: tmp
    if (k <= (-1.1598924007845897d-155)) then
        tmp = 2.0d0 * (((l / k) / ((sin(k) ** 2.0d0) * (t / cos(k)))) / (k / l))
    else if (k <= 6.310209504498685d-147) then
        tmp = 2.0d0 * (((cos(k) * (((l / k) ** 2.0d0) / t)) / sin(k)) / sin(k))
    else
        tmp = 2.0d0 * (((l / k) / (t / (cos(k) * (sin(k) ** (-2.0d0))))) / (k / 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 tmp;
	if (k <= -1.1598924007845897e-155) {
		tmp = 2.0 * (((l / k) / (Math.pow(Math.sin(k), 2.0) * (t / Math.cos(k)))) / (k / l));
	} else if (k <= 6.310209504498685e-147) {
		tmp = 2.0 * (((Math.cos(k) * (Math.pow((l / k), 2.0) / t)) / Math.sin(k)) / Math.sin(k));
	} else {
		tmp = 2.0 * (((l / k) / (t / (Math.cos(k) * Math.pow(Math.sin(k), -2.0)))) / (k / 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):
	tmp = 0
	if k <= -1.1598924007845897e-155:
		tmp = 2.0 * (((l / k) / (math.pow(math.sin(k), 2.0) * (t / math.cos(k)))) / (k / l))
	elif k <= 6.310209504498685e-147:
		tmp = 2.0 * (((math.cos(k) * (math.pow((l / k), 2.0) / t)) / math.sin(k)) / math.sin(k))
	else:
		tmp = 2.0 * (((l / k) / (t / (math.cos(k) * math.pow(math.sin(k), -2.0)))) / (k / 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)
	tmp = 0.0
	if (k <= -1.1598924007845897e-155)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64((sin(k) ^ 2.0) * Float64(t / cos(k)))) / Float64(k / l)));
	elseif (k <= 6.310209504498685e-147)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64((Float64(l / k) ^ 2.0) / t)) / sin(k)) / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / Float64(t / Float64(cos(k) * (sin(k) ^ -2.0)))) / Float64(k / 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)
	tmp = 0.0;
	if (k <= -1.1598924007845897e-155)
		tmp = 2.0 * (((l / k) / ((sin(k) ^ 2.0) * (t / cos(k)))) / (k / l));
	elseif (k <= 6.310209504498685e-147)
		tmp = 2.0 * (((cos(k) * (((l / k) ^ 2.0) / t)) / sin(k)) / sin(k));
	else
		tmp = 2.0 * (((l / k) / (t / (cos(k) * (sin(k) ^ -2.0)))) / (k / 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_] := If[LessEqual[k, -1.1598924007845897e-155], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.310209504498685e-147], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[(t / N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $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 k < -1.1598924007845897e-155

   1. Initial program 47.5

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

   2. Simplified 40.0

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

   3. Taylor expanded in t around 0 21.6

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

   4. Simplified 5.5

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

   5. Applied egg-rr 1.3

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

   6. Applied egg-rr 1.9

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}}}{\frac{k}{\ell}} \]

   7. Taylor expanded in t around 0 1.9

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

   8. Simplified 1.9

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

   if -1.1598924007845897e-155 < k < 6.310209504498685e-147

   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

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

   3. Taylor expanded in t around 0 63.8

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

   4. Simplified 60.9

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

   5. Applied egg-rr 16.9

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

   if 6.310209504498685e-147 < k

   1. Initial program 47.9

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

   2. Simplified 40.3

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

   3. Taylor expanded in t around 0 21.3

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

   4. Simplified 5.9

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

   5. Applied egg-rr 1.2

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

   6. Applied egg-rr 1.5

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}}}{\frac{k}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.1598924007845897 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{{\sin k}^{2} \cdot \frac{t}{\cos k}}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 6.310209504498685 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{\frac{t}{\cos k \cdot {\sin k}^{-2}}}}{\frac{k}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))