Toniolo and Linder, Equation (10-)

Average Error: 26.0% → 98.8%

Time: 27.9s

Precision: binary64

Cost: 20489.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -550000000 \lor \neg \left(k \leq 4 \cdot 10^{-75}\right):\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\frac{\ell}{\frac{k}{\cos k}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

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 -550000000.0) (not (<= k 4e-75)))
   (* 2.0 (* (/ (/ l k) (pow (sin k) 2.0)) (/ (/ l (/ k (cos k))) t)))
   (* 2.0 (/ (/ (/ l k) k) (* t (* k (/ 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 <= -550000000.0) || !(k <= 4e-75)) {
		tmp = 2.0 * (((l / k) / pow(sin(k), 2.0)) * ((l / (k / cos(k))) / t));
	} else {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (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 <= (-550000000.0d0)) .or. (.not. (k <= 4d-75))) then
        tmp = 2.0d0 * (((l / k) / (sin(k) ** 2.0d0)) * ((l / (k / cos(k))) / t))
    else
        tmp = 2.0d0 * (((l / k) / k) / (t * (k * (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 <= -550000000.0) || !(k <= 4e-75)) {
		tmp = 2.0 * (((l / k) / Math.pow(Math.sin(k), 2.0)) * ((l / (k / Math.cos(k))) / t));
	} else {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (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 <= -550000000.0) or not (k <= 4e-75):
		tmp = 2.0 * (((l / k) / math.pow(math.sin(k), 2.0)) * ((l / (k / math.cos(k))) / t))
	else:
		tmp = 2.0 * (((l / k) / k) / (t * (k * (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 <= -550000000.0) || !(k <= 4e-75))
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / (sin(k) ^ 2.0)) * Float64(Float64(l / Float64(k / cos(k))) / t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * Float64(k * 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 <= -550000000.0) || ~((k <= 4e-75)))
		tmp = 2.0 * (((l / k) / (sin(k) ^ 2.0)) * ((l / (k / cos(k))) / t));
	else
		tmp = 2.0 * (((l / k) / k) / (t * (k * (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[Or[LessEqual[k, -550000000.0], N[Not[LessEqual[k, 4e-75]], $MachinePrecision]], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.5e8 or 3.9999999999999998e-75 < k

   1. Initial program 29.8

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

   2. Simplified 44.6

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

      [Start] 29.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 29.7	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 29.7	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 29.7	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 31.7	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 31.7	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 31.7	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 44.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 44.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 44.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around inf 69.4

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

   4. Simplified 69.4

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

      [Start] 69.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 69.4	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 69.4	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 69.4	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 69.4	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 93.2

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

   6. Taylor expanded in k around inf 92.5

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

   7. Simplified 99.3

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

      [Start] 92.5	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{k \cdot \left({\sin k}^{2} \cdot t\right)}{\ell}} \]
      associate-*r* [=>] 92.5	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{\color{blue}{\left(k \cdot {\sin k}^{2}\right) \cdot t}}{\ell}} \]
      associate-*l/ [<=] 99.3	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\color{blue}{\frac{k \cdot {\sin k}^{2}}{\ell} \cdot t}} \]

   8. Applied egg-rr 99.3

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

   if -5.5e8 < k < 3.9999999999999998e-75

   1. Initial program 3.5

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

   2. Simplified 19.6

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

      [Start] 3.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 3.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 3.7	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 3.7	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 5.4	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 5.4	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 5.4	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 4.8	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 4.8	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 4.8	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 21.6

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 26.6

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 21.6	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 21.6	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 26.6	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 64.0

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

   6. Applied egg-rr 95.7

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

4. Final simplification 98.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -550000000 \lor \neg \left(k \leq 4 \cdot 10^{-75}\right):\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\frac{\ell}{\frac{k}{\cos k}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	78.9%
Cost	14409.00

\[\begin{array}{l} \mathbf{if}\;k \leq -550000000 \lor \neg \left(k \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \frac{2 \cdot \frac{\ell}{t}}{1 - \cos \left(k + k\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	93.5%
Cost	14409.00

\[\begin{array}{l} \mathbf{if}\;k \leq -550000000 \lor \neg \left(k \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	98.5%
Cost	14409.00

\[\begin{array}{l} \mathbf{if}\;k \leq -550000000 \lor \neg \left(k \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{t \cdot \frac{k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	59.6%
Cost	960.00

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

Alternative 5
Error	60.3%
Cost	960.00

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

Alternative 6
Error	61.4%
Cost	960.00

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

Alternative 7
Error	63.5%
Cost	960.00

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

Alternative 8
Error	64.6%
Cost	960.00

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

Reproduce

herbie shell --seed 2023097 
(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))))