Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 13.9

Time: 25.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := k \cdot \left(\sin k \cdot \sqrt{t}\right)\\ t_3 := \cos k \cdot \ell\\ t_4 := \left(k \cdot k\right) \cdot \frac{t \cdot t_1}{t_3}\\ \mathbf{if}\;t \leq -4.705346351463268 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t}{t_3} \cdot t_1}{\ell}}\\ \mathbf{elif}\;t \leq -5.733002091172997 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{t_4}}}\\ \mathbf{elif}\;t \leq 1.3030292772025043 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t_1 \cdot \left(t \cdot \frac{1}{t_3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.0228613744328519 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\cos k} \cdot \frac{t_2}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_4}{\ell}}\\ \end{array} \]

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
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2 (* k (* (sin k) (sqrt t))))
        (t_3 (* (cos k) l))
        (t_4 (* (* k k) (/ (* t t_1) t_3))))
   (if (<= t -4.705346351463268e+166)
     (/ 2.0 (* (* k k) (/ (* (/ t t_3) t_1) l)))
     (if (<= t -5.733002091172997e-223)
       (/ 2.0 (/ 1.0 (/ l t_4)))
       (if (<= t 1.3030292772025043e-243)
         (/ 2.0 (* (* k k) (/ (* t_1 (* t (/ 1.0 t_3))) l)))
         (if (<= t 1.0228613744328519e-44)
           (/ 2.0 (* (/ t_2 (cos k)) (/ t_2 (* l l))))
           (/ 2.0 (/ t_4 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 t_1 = pow(sin(k), 2.0);
	double t_2 = k * (sin(k) * sqrt(t));
	double t_3 = cos(k) * l;
	double t_4 = (k * k) * ((t * t_1) / t_3);
	double tmp;
	if (t <= -4.705346351463268e+166) {
		tmp = 2.0 / ((k * k) * (((t / t_3) * t_1) / l));
	} else if (t <= -5.733002091172997e-223) {
		tmp = 2.0 / (1.0 / (l / t_4));
	} else if (t <= 1.3030292772025043e-243) {
		tmp = 2.0 / ((k * k) * ((t_1 * (t * (1.0 / t_3))) / l));
	} else if (t <= 1.0228613744328519e-44) {
		tmp = 2.0 / ((t_2 / cos(k)) * (t_2 / (l * l)));
	} else {
		tmp = 2.0 / (t_4 / 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = k * (sin(k) * sqrt(t))
    t_3 = cos(k) * l
    t_4 = (k * k) * ((t * t_1) / t_3)
    if (t <= (-4.705346351463268d+166)) then
        tmp = 2.0d0 / ((k * k) * (((t / t_3) * t_1) / l))
    else if (t <= (-5.733002091172997d-223)) then
        tmp = 2.0d0 / (1.0d0 / (l / t_4))
    else if (t <= 1.3030292772025043d-243) then
        tmp = 2.0d0 / ((k * k) * ((t_1 * (t * (1.0d0 / t_3))) / l))
    else if (t <= 1.0228613744328519d-44) then
        tmp = 2.0d0 / ((t_2 / cos(k)) * (t_2 / (l * l)))
    else
        tmp = 2.0d0 / (t_4 / 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 t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = k * (Math.sin(k) * Math.sqrt(t));
	double t_3 = Math.cos(k) * l;
	double t_4 = (k * k) * ((t * t_1) / t_3);
	double tmp;
	if (t <= -4.705346351463268e+166) {
		tmp = 2.0 / ((k * k) * (((t / t_3) * t_1) / l));
	} else if (t <= -5.733002091172997e-223) {
		tmp = 2.0 / (1.0 / (l / t_4));
	} else if (t <= 1.3030292772025043e-243) {
		tmp = 2.0 / ((k * k) * ((t_1 * (t * (1.0 / t_3))) / l));
	} else if (t <= 1.0228613744328519e-44) {
		tmp = 2.0 / ((t_2 / Math.cos(k)) * (t_2 / (l * l)));
	} else {
		tmp = 2.0 / (t_4 / 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):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = k * (math.sin(k) * math.sqrt(t))
	t_3 = math.cos(k) * l
	t_4 = (k * k) * ((t * t_1) / t_3)
	tmp = 0
	if t <= -4.705346351463268e+166:
		tmp = 2.0 / ((k * k) * (((t / t_3) * t_1) / l))
	elif t <= -5.733002091172997e-223:
		tmp = 2.0 / (1.0 / (l / t_4))
	elif t <= 1.3030292772025043e-243:
		tmp = 2.0 / ((k * k) * ((t_1 * (t * (1.0 / t_3))) / l))
	elif t <= 1.0228613744328519e-44:
		tmp = 2.0 / ((t_2 / math.cos(k)) * (t_2 / (l * l)))
	else:
		tmp = 2.0 / (t_4 / 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)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(k * Float64(sin(k) * sqrt(t)))
	t_3 = Float64(cos(k) * l)
	t_4 = Float64(Float64(k * k) * Float64(Float64(t * t_1) / t_3))
	tmp = 0.0
	if (t <= -4.705346351463268e+166)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t / t_3) * t_1) / l)));
	elseif (t <= -5.733002091172997e-223)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / t_4)));
	elseif (t <= 1.3030292772025043e-243)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_1 * Float64(t * Float64(1.0 / t_3))) / l)));
	elseif (t <= 1.0228613744328519e-44)
		tmp = Float64(2.0 / Float64(Float64(t_2 / cos(k)) * Float64(t_2 / Float64(l * l))));
	else
		tmp = Float64(2.0 / Float64(t_4 / 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)
	t_1 = sin(k) ^ 2.0;
	t_2 = k * (sin(k) * sqrt(t));
	t_3 = cos(k) * l;
	t_4 = (k * k) * ((t * t_1) / t_3);
	tmp = 0.0;
	if (t <= -4.705346351463268e+166)
		tmp = 2.0 / ((k * k) * (((t / t_3) * t_1) / l));
	elseif (t <= -5.733002091172997e-223)
		tmp = 2.0 / (1.0 / (l / t_4));
	elseif (t <= 1.3030292772025043e-243)
		tmp = 2.0 / ((k * k) * ((t_1 * (t * (1.0 / t_3))) / l));
	elseif (t <= 1.0228613744328519e-44)
		tmp = 2.0 / ((t_2 / cos(k)) * (t_2 / (l * l)));
	else
		tmp = 2.0 / (t_4 / 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_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$4 = N[(N[(k * k), $MachinePrecision] * N[(N[(t * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.705346351463268e+166], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.733002091172997e-223], N[(2.0 / N[(1.0 / N[(l / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3030292772025043e-243], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$1 * N[(t * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.0228613744328519e-44], N[(2.0 / N[(N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$4 / l), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if t < -4.7053463514632678e166

   1. Initial program 56.8

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

   2. Simplified 42.4

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

   3. Taylor expanded in t around 0 22.5

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

   4. Applied egg-rr 20.7

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

   5. Applied egg-rr 12.5

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

   6. Applied egg-rr 14.6

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

   if -4.7053463514632678e166 < t < -5.7330020911729971e-223

   1. Initial program 41.8

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

   2. Simplified 35.8

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

   3. Taylor expanded in t around 0 22.8

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

   4. Applied egg-rr 21.6

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

   5. Applied egg-rr 17.4

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

   6. Applied egg-rr 13.7

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

   if -5.7330020911729971e-223 < t < 1.30302927720250425e-243

   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

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

   3. Taylor expanded in t around 0 27.3

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

   4. Applied egg-rr 31.5

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

   5. Applied egg-rr 27.8

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

   6. Applied egg-rr 21.4

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

   if 1.30302927720250425e-243 < t < 1.02286137443285191e-44

   1. Initial program 53.3

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

   2. Simplified 53.0

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

   3. Taylor expanded in t around 0 24.0

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

   4. Applied egg-rr 15.7

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

   if 1.02286137443285191e-44 < t

   1. Initial program 43.8

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

   2. Simplified 31.6

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

   3. Taylor expanded in t around 0 21.2

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

   4. Applied egg-rr 18.9

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

   5. Applied egg-rr 13.5

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

   6. Applied egg-rr 11.1

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot \ell}}{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.705346351463268 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t}{\cos k \cdot \ell} \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq -5.733002091172997 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot \ell}}}}\\ \mathbf{elif}\;t \leq 1.3030292772025043 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{{\sin k}^{2} \cdot \left(t \cdot \frac{1}{\cos k \cdot \ell}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.0228613744328519 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\cos k} \cdot \frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot \ell}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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))))