Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 0.8

Time: 27.9s

Precision: binary64

Cost: 14025

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -85000 \lor \neg \left(k \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{k}{\ell} \cdot t} \cdot \frac{\frac{\ell}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(k \cdot -0.6666666666666666 + 2 \cdot \frac{1}{k}\right)}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

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 (or (<= k -85000.0) (not (<= k 1.8e-120)))
   (* (/ (/ 2.0 (tan k)) (* (/ k l) t)) (/ (/ l k) (sin k)))
   (/
    (* (/ l k) (+ (* k -0.6666666666666666) (* 2.0 (/ 1.0 k))))
    (* (* (sin k) (/ k l)) t))))

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 <= -85000.0) || !(k <= 1.8e-120)) {
		tmp = ((2.0 / tan(k)) / ((k / l) * t)) * ((l / k) / sin(k));
	} else {
		tmp = ((l / k) * ((k * -0.6666666666666666) + (2.0 * (1.0 / k)))) / ((sin(k) * (k / l)) * t);
	}
	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 <= (-85000.0d0)) .or. (.not. (k <= 1.8d-120))) then
        tmp = ((2.0d0 / tan(k)) / ((k / l) * t)) * ((l / k) / sin(k))
    else
        tmp = ((l / k) * ((k * (-0.6666666666666666d0)) + (2.0d0 * (1.0d0 / k)))) / ((sin(k) * (k / l)) * t)
    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 <= -85000.0) || !(k <= 1.8e-120)) {
		tmp = ((2.0 / Math.tan(k)) / ((k / l) * t)) * ((l / k) / Math.sin(k));
	} else {
		tmp = ((l / k) * ((k * -0.6666666666666666) + (2.0 * (1.0 / k)))) / ((Math.sin(k) * (k / l)) * t);
	}
	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 <= -85000.0) or not (k <= 1.8e-120):
		tmp = ((2.0 / math.tan(k)) / ((k / l) * t)) * ((l / k) / math.sin(k))
	else:
		tmp = ((l / k) * ((k * -0.6666666666666666) + (2.0 * (1.0 / k)))) / ((math.sin(k) * (k / l)) * t)
	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 <= -85000.0) || !(k <= 1.8e-120))
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(Float64(k / l) * t)) * Float64(Float64(l / k) / sin(k)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(k * -0.6666666666666666) + Float64(2.0 * Float64(1.0 / k)))) / Float64(Float64(sin(k) * Float64(k / l)) * t));
	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 <= -85000.0) || ~((k <= 1.8e-120)))
		tmp = ((2.0 / tan(k)) / ((k / l) * t)) * ((l / k) / sin(k));
	else
		tmp = ((l / k) * ((k * -0.6666666666666666) + (2.0 * (1.0 / k)))) / ((sin(k) * (k / l)) * t);
	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[Or[LessEqual[k, -85000.0], N[Not[LessEqual[k, 1.8e-120]], $MachinePrecision]], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(k * -0.6666666666666666), $MachinePrecision] + N[(2.0 * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -85000 or 1.8000000000000001e-120 < k

   1. Initial program 46.0

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

   2. Simplified 37.8

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 46.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 46.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 46.0	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 46.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 37.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 37.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 19.9

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

   4. Simplified 13.5

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

      [Start] 19.9	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 20.1	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 20.1	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 13.5	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 13.5	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 13.5	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 4.5

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

   6. Taylor expanded in t around 0 4.3

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

   7. Simplified 0.5

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{t \cdot \frac{k}{\ell}}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
      Proof

      [Start] 4.3	\[ \frac{\frac{2}{\tan k}}{\frac{k \cdot t}{\ell}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
      associate-*l/ [<=] 0.5	\[ \frac{\frac{2}{\tan k}}{\color{blue}{\frac{k}{\ell} \cdot t}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
      *-commutative [<=] 0.5	\[ \frac{\frac{2}{\tan k}}{\color{blue}{t \cdot \frac{k}{\ell}}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]

   if -85000 < k < 1.8000000000000001e-120

   1. Initial program 61.9

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

   2. Simplified 56.5

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 61.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 61.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 61.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 61.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 56.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 56.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 38.0

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

   4. Simplified 33.0

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

      [Start] 38.0	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 42.7	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 42.7	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 33.0	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 33.0	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 33.0	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 24.0

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

   6. Simplified 0.5

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

      [Start] 24.0	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k \cdot \sin k} \cdot \frac{\ell}{k} \]
      *-commutative [=>] 24.0	\[ \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k \cdot \sin k}} \]
      associate-/l* [=>] 24.0	\[ \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\frac{k \cdot \sin k}{\frac{\ell}{t}}}} \]
      associate-*r/ [=>] 20.7	\[ \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{k \cdot \sin k}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 13.7	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot t}} \]
      *-commutative [=>] 13.7	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\sin k \cdot k}}{\ell} \cdot t} \]
      *-rgt-identity [<=] 13.7	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{\sin k \cdot \color{blue}{\left(k \cdot 1\right)}}{\ell} \cdot t} \]
      associate-*r/ [<=] 0.5	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot \frac{k \cdot 1}{\ell}\right)} \cdot t} \]
      *-rgt-identity [=>] 0.5	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\left(\sin k \cdot \frac{\color{blue}{k}}{\ell}\right) \cdot t} \]

   7. Taylor expanded in k around 0 3.1

      \[\leadsto \frac{\frac{\ell}{k} \cdot \color{blue}{\left(-0.6666666666666666 \cdot k + 2 \cdot \frac{1}{k}\right)}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -85000 \lor \neg \left(k \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{k}{\ell} \cdot t} \cdot \frac{\frac{\ell}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(k \cdot -0.6666666666666666 + 2 \cdot \frac{1}{k}\right)}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.5
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -85000 \lor \neg \left(k \leq 1.4 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\sin k} \cdot \left(\ell \cdot \frac{\frac{2}{k \cdot \tan k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)}{k}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13760

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

Alternative 3
Error	0.3
Cost	13760

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

Alternative 4
Error	22.8
Cost	7360

\[\frac{2 \cdot \frac{\frac{\ell}{k}}{k}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t} \]

Alternative 5
Error	22.9
Cost	7360

\[\frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{t \cdot \frac{k}{\frac{\ell}{k}}} \]

Alternative 6
Error	26.1
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \frac{2}{t} \cdot \left(t_1 \cdot t_1\right) \end{array} \]

Alternative 7
Error	26.1
Cost	960

\[\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{\frac{k \cdot k}{\frac{\ell}{k}}} \]

Alternative 8
Error	23.0
Cost	960

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \frac{2}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Reproduce

herbie shell --seed 2023057 
(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))))