Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 6.9

Time: 22.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sin k \cdot \sqrt{t}\\ t_2 := 2 \cdot \cos k\\ t_3 := {\left(\frac{\ell}{k}\right)}^{2}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{t_3}{\frac{t \cdot t_4}{t_2}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\frac{k}{\ell}}}{\frac{t}{\frac{\cos k}{k}}}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_3 \cdot t_2}{t_1}}{t_1}\\ \end{array} \]

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 (* (sin k) (sqrt t)))
        (t_2 (* 2.0 (cos k)))
        (t_3 (pow (/ l k) 2.0))
        (t_4 (pow (sin k) 2.0)))
   (if (<= t -3.5e-90)
     (/ t_3 (/ (* t t_4) t_2))
     (if (<= t 3.8e-105)
       (* 2.0 (/ (/ (/ l (/ k l)) (/ t (/ (cos k) k))) t_4))
       (/ (/ (* t_3 t_2) t_1) t_1)))))

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 = sin(k) * sqrt(t);
	double t_2 = 2.0 * cos(k);
	double t_3 = pow((l / k), 2.0);
	double t_4 = pow(sin(k), 2.0);
	double tmp;
	if (t <= -3.5e-90) {
		tmp = t_3 / ((t * t_4) / t_2);
	} else if (t <= 3.8e-105) {
		tmp = 2.0 * (((l / (k / l)) / (t / (cos(k) / k))) / t_4);
	} else {
		tmp = ((t_3 * t_2) / t_1) / t_1;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sin(k) * sqrt(t)
    t_2 = 2.0d0 * cos(k)
    t_3 = (l / k) ** 2.0d0
    t_4 = sin(k) ** 2.0d0
    if (t <= (-3.5d-90)) then
        tmp = t_3 / ((t * t_4) / t_2)
    else if (t <= 3.8d-105) then
        tmp = 2.0d0 * (((l / (k / l)) / (t / (cos(k) / k))) / t_4)
    else
        tmp = ((t_3 * t_2) / t_1) / t_1
    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.sin(k) * Math.sqrt(t);
	double t_2 = 2.0 * Math.cos(k);
	double t_3 = Math.pow((l / k), 2.0);
	double t_4 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= -3.5e-90) {
		tmp = t_3 / ((t * t_4) / t_2);
	} else if (t <= 3.8e-105) {
		tmp = 2.0 * (((l / (k / l)) / (t / (Math.cos(k) / k))) / t_4);
	} else {
		tmp = ((t_3 * t_2) / t_1) / t_1;
	}
	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.sin(k) * math.sqrt(t)
	t_2 = 2.0 * math.cos(k)
	t_3 = math.pow((l / k), 2.0)
	t_4 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= -3.5e-90:
		tmp = t_3 / ((t * t_4) / t_2)
	elif t <= 3.8e-105:
		tmp = 2.0 * (((l / (k / l)) / (t / (math.cos(k) / k))) / t_4)
	else:
		tmp = ((t_3 * t_2) / t_1) / t_1
	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(sin(k) * sqrt(t))
	t_2 = Float64(2.0 * cos(k))
	t_3 = Float64(l / k) ^ 2.0
	t_4 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= -3.5e-90)
		tmp = Float64(t_3 / Float64(Float64(t * t_4) / t_2));
	elseif (t <= 3.8e-105)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k / l)) / Float64(t / Float64(cos(k) / k))) / t_4));
	else
		tmp = Float64(Float64(Float64(t_3 * t_2) / t_1) / t_1);
	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 = sin(k) * sqrt(t);
	t_2 = 2.0 * cos(k);
	t_3 = (l / k) ^ 2.0;
	t_4 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t <= -3.5e-90)
		tmp = t_3 / ((t * t_4) / t_2);
	elseif (t <= 3.8e-105)
		tmp = 2.0 * (((l / (k / l)) / (t / (cos(k) / k))) / t_4);
	else
		tmp = ((t_3 * t_2) / t_1) / t_1;
	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[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -3.5e-90], N[(t$95$3 / N[(N[(t * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-105], N[(2.0 * N[(N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(t / N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -3.4999999999999999e-90

   1. Initial program 43.1

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

   2. Simplified 32.4

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

   3. Taylor expanded in t around 0 20.5

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

   4. Simplified 18.3

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

   5. Taylor expanded in k around inf 20.5

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

   6. Simplified 12.4

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

   7. Applied egg-rr 6.9

      \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\cos k \cdot 2\right)}}} \]

   8. Applied egg-rr 6.7

      \[\leadsto \color{blue}{{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{t \cdot {\sin k}^{2}}{2 \cdot \cos k}}\right)}^{1}} \]

   if -3.4999999999999999e-90 < t < 3.7999999999999998e-105

   1. Initial program 62.0

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

   2. Simplified 62.1

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

   3. Taylor expanded in t around 0 26.7

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

   4. Simplified 29.6

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

   5. Taylor expanded in k around inf 26.7

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

   6. Simplified 27.4

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

   7. Taylor expanded in l around 0 26.7

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

   8. Simplified 10.6

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

   if 3.7999999999999998e-105 < t

   1. Initial program 40.7

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

   2. Simplified 30.5

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

   3. Taylor expanded in t around 0 21.0

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

   4. Simplified 19.4

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

   5. Taylor expanded in k around inf 21.0

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

   6. Simplified 13.0

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

   7. Applied egg-rr 3.8

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{\frac{t \cdot {\sin k}^{2}}{2 \cdot \cos k}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{\frac{k}{\ell}}}{\frac{t}{\frac{\cos k}{k}}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(2 \cdot \cos k\right)}{\sin k \cdot \sqrt{t}}}{\sin k \cdot \sqrt{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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))))