Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 10.2

Time: 30.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{{\sin k}^{2}}{\ell}\\ t_2 := \frac{2}{\left(k \cdot \left(\frac{k}{\cos k} \cdot \frac{t}{\ell}\right)\right) \cdot t_1}\\ \mathbf{if}\;k \leq -1.0296543156802884 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.2685856286452325 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (pow (sin k) 2.0) l))
        (t_2 (/ 2.0 (* (* k (* (/ k (cos k)) (/ t l))) t_1))))
   (if (<= k -1.0296543156802884e-7)
     t_2
     (if (<= k 2.2685856286452325e-39)
       (/ 2.0 (* t_1 (* (* t (/ (pow k 2.0) (cos k))) (/ 1.0 l))))
       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 = pow(sin(k), 2.0) / l;
	double t_2 = 2.0 / ((k * ((k / cos(k)) * (t / l))) * t_1);
	double tmp;
	if (k <= -1.0296543156802884e-7) {
		tmp = t_2;
	} else if (k <= 2.2685856286452325e-39) {
		tmp = 2.0 / (t_1 * ((t * (pow(k, 2.0) / cos(k))) * (1.0 / l)));
	} 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 = (sin(k) ** 2.0d0) / l
    t_2 = 2.0d0 / ((k * ((k / cos(k)) * (t / l))) * t_1)
    if (k <= (-1.0296543156802884d-7)) then
        tmp = t_2
    else if (k <= 2.2685856286452325d-39) then
        tmp = 2.0d0 / (t_1 * ((t * ((k ** 2.0d0) / cos(k))) * (1.0d0 / l)))
    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.pow(Math.sin(k), 2.0) / l;
	double t_2 = 2.0 / ((k * ((k / Math.cos(k)) * (t / l))) * t_1);
	double tmp;
	if (k <= -1.0296543156802884e-7) {
		tmp = t_2;
	} else if (k <= 2.2685856286452325e-39) {
		tmp = 2.0 / (t_1 * ((t * (Math.pow(k, 2.0) / Math.cos(k))) * (1.0 / l)));
	} 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.pow(math.sin(k), 2.0) / l
	t_2 = 2.0 / ((k * ((k / math.cos(k)) * (t / l))) * t_1)
	tmp = 0
	if k <= -1.0296543156802884e-7:
		tmp = t_2
	elif k <= 2.2685856286452325e-39:
		tmp = 2.0 / (t_1 * ((t * (math.pow(k, 2.0) / math.cos(k))) * (1.0 / l)))
	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((sin(k) ^ 2.0) / l)
	t_2 = Float64(2.0 / Float64(Float64(k * Float64(Float64(k / cos(k)) * Float64(t / l))) * t_1))
	tmp = 0.0
	if (k <= -1.0296543156802884e-7)
		tmp = t_2;
	elseif (k <= 2.2685856286452325e-39)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t * Float64((k ^ 2.0) / cos(k))) * Float64(1.0 / l))));
	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 = (sin(k) ^ 2.0) / l;
	t_2 = 2.0 / ((k * ((k / cos(k)) * (t / l))) * t_1);
	tmp = 0.0;
	if (k <= -1.0296543156802884e-7)
		tmp = t_2;
	elseif (k <= 2.2685856286452325e-39)
		tmp = 2.0 / (t_1 * ((t * ((k ^ 2.0) / cos(k))) * (1.0 / l)));
	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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(k * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.0296543156802884e-7], t$95$2, If[LessEqual[k, 2.2685856286452325e-39], N[(2.0 / N[(t$95$1 * N[(N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $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 < -1.0296543156802884e-7 or 2.2685856286452325e-39 < k

   1. Initial program 44.8

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

   2. Simplified 36.9

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

   3. Taylor expanded in t around 0 19.2

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

   4. Applied times-frac_binary64 19.1

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

   5. Applied unpow2_binary64 19.1

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

   6. Applied times-frac_binary64 14.8

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

   7. Applied associate-*r*_binary64 12.7

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

   8. Applied *-un-lft-identity_binary64 12.7

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

   9. Applied unpow2_binary64 12.7

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

   10. Applied times-frac_binary64 12.7

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

   11. Applied associate-*l*_binary64 8.1

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

   if -1.0296543156802884e-7 < k < 2.2685856286452325e-39

   1. Initial program 62.2

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

   2. Simplified 56.0

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

   3. Taylor expanded in t around 0 37.6

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

   4. Applied times-frac_binary64 33.1

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

   5. Applied unpow2_binary64 33.1

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

   6. Applied times-frac_binary64 21.2

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

   7. Applied associate-*r*_binary64 19.2

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

   8. Applied div-inv_binary64 19.2

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

   9. Applied associate-*r*_binary64 18.5

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.0296543156802884 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\frac{k}{\cos k} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 2.2685856286452325 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \left(\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\frac{k}{\cos k} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{{\sin k}^{2}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))