Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 1.6

Time: 26.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\cos k}{k} \cdot \ell\\ t_2 := 2 \cdot \frac{t_1}{k \cdot \frac{t}{\frac{1}{\sin k} \cdot \frac{\ell}{\sin k}}}\\ \mathbf{if}\;t \leq -1.1128609500028873 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.903013830898229 \cdot 10^{+63}:\\ \;\;\;\;2 \cdot \frac{t_1}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* (/ (cos k) k) l))
        (t_2 (* 2.0 (/ t_1 (* k (/ t (* (/ 1.0 (sin k)) (/ l (sin k)))))))))
   (if (<= t -1.1128609500028873e+63)
     t_2
     (if (<= t 5.903013830898229e+63)
       (* 2.0 (/ t_1 (/ (* t k) (/ l (pow (sin k) 2.0)))))
       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 = (cos(k) / k) * l;
	double t_2 = 2.0 * (t_1 / (k * (t / ((1.0 / sin(k)) * (l / sin(k))))));
	double tmp;
	if (t <= -1.1128609500028873e+63) {
		tmp = t_2;
	} else if (t <= 5.903013830898229e+63) {
		tmp = 2.0 * (t_1 / ((t * k) / (l / pow(sin(k), 2.0))));
	} 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 = (cos(k) / k) * l
    t_2 = 2.0d0 * (t_1 / (k * (t / ((1.0d0 / sin(k)) * (l / sin(k))))))
    if (t <= (-1.1128609500028873d+63)) then
        tmp = t_2
    else if (t <= 5.903013830898229d+63) then
        tmp = 2.0d0 * (t_1 / ((t * k) / (l / (sin(k) ** 2.0d0))))
    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.cos(k) / k) * l;
	double t_2 = 2.0 * (t_1 / (k * (t / ((1.0 / Math.sin(k)) * (l / Math.sin(k))))));
	double tmp;
	if (t <= -1.1128609500028873e+63) {
		tmp = t_2;
	} else if (t <= 5.903013830898229e+63) {
		tmp = 2.0 * (t_1 / ((t * k) / (l / Math.pow(Math.sin(k), 2.0))));
	} 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.cos(k) / k) * l
	t_2 = 2.0 * (t_1 / (k * (t / ((1.0 / math.sin(k)) * (l / math.sin(k))))))
	tmp = 0
	if t <= -1.1128609500028873e+63:
		tmp = t_2
	elif t <= 5.903013830898229e+63:
		tmp = 2.0 * (t_1 / ((t * k) / (l / math.pow(math.sin(k), 2.0))))
	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(Float64(cos(k) / k) * l)
	t_2 = Float64(2.0 * Float64(t_1 / Float64(k * Float64(t / Float64(Float64(1.0 / sin(k)) * Float64(l / sin(k)))))))
	tmp = 0.0
	if (t <= -1.1128609500028873e+63)
		tmp = t_2;
	elseif (t <= 5.903013830898229e+63)
		tmp = Float64(2.0 * Float64(t_1 / Float64(Float64(t * k) / Float64(l / (sin(k) ^ 2.0)))));
	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 = (cos(k) / k) * l;
	t_2 = 2.0 * (t_1 / (k * (t / ((1.0 / sin(k)) * (l / sin(k))))));
	tmp = 0.0;
	if (t <= -1.1128609500028873e+63)
		tmp = t_2;
	elseif (t <= 5.903013830898229e+63)
		tmp = 2.0 * (t_1 / ((t * k) / (l / (sin(k) ^ 2.0))));
	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[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t$95$1 / N[(k * N[(t / N[(N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.1128609500028873e+63], t$95$2, If[LessEqual[t, 5.903013830898229e+63], N[(2.0 * N[(t$95$1 / N[(N[(t * k), $MachinePrecision] / N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $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 t < -1.1128609500028873e63 or 5.9030138308982288e63 < t

   1. Initial program 49.8

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

   2. Simplified 35.7

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

   3. Taylor expanded in t around 0 20.9

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

   4. Applied egg-rr 18.2

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

   5. Applied egg-rr 3.8

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

   6. Applied egg-rr 0.4

      \[\leadsto 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{k \cdot \frac{t}{\color{blue}{\frac{1}{\sin k} \cdot \frac{\ell}{\sin k}}}} \]

   if -1.1128609500028873e63 < t < 5.9030138308982288e63

   1. Initial program 46.5

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

   2. Simplified 43.8

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

   3. Taylor expanded in t around 0 24.0

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

   4. Applied egg-rr 24.7

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

   5. Applied egg-rr 7.2

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

   6. Applied egg-rr 2.7

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1128609500028873 \cdot 10^{+63}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{k \cdot \frac{t}{\frac{1}{\sin k} \cdot \frac{\ell}{\sin k}}}\\ \mathbf{elif}\;t \leq 5.903013830898229 \cdot 10^{+63}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{k \cdot \frac{t}{\frac{1}{\sin k} \cdot \frac{\ell}{\sin k}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))