Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 6.3

Time: 28.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \cos k\\ t_2 := k \cdot \left(\sin k \cdot \sqrt{t}\right)\\ \mathbf{if}\;t \leq 3.4056655695663573 \cdot 10^{-273}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell} \cdot \frac{k}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\ell} \cdot \frac{t_2}{t_1}}\\ \end{array} \]

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 (* l (cos k))) (t_2 (* k (* (sin k) (sqrt t)))))
   (if (<= t 3.4056655695663573e-273)
     (/ 2.0 (* (/ (* k (* t (pow (sin k) 2.0))) l) (/ k t_1)))
     (/ 2.0 (* (/ t_2 l) (/ t_2 t_1))))))

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 = l * cos(k);
	double t_2 = k * (sin(k) * sqrt(t));
	double tmp;
	if (t <= 3.4056655695663573e-273) {
		tmp = 2.0 / (((k * (t * pow(sin(k), 2.0))) / l) * (k / t_1));
	} else {
		tmp = 2.0 / ((t_2 / l) * (t_2 / t_1));
	}
	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 = l * cos(k)
    t_2 = k * (sin(k) * sqrt(t))
    if (t <= 3.4056655695663573d-273) then
        tmp = 2.0d0 / (((k * (t * (sin(k) ** 2.0d0))) / l) * (k / t_1))
    else
        tmp = 2.0d0 / ((t_2 / l) * (t_2 / t_1))
    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 = l * Math.cos(k);
	double t_2 = k * (Math.sin(k) * Math.sqrt(t));
	double tmp;
	if (t <= 3.4056655695663573e-273) {
		tmp = 2.0 / (((k * (t * Math.pow(Math.sin(k), 2.0))) / l) * (k / t_1));
	} else {
		tmp = 2.0 / ((t_2 / l) * (t_2 / t_1));
	}
	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 = l * math.cos(k)
	t_2 = k * (math.sin(k) * math.sqrt(t))
	tmp = 0
	if t <= 3.4056655695663573e-273:
		tmp = 2.0 / (((k * (t * math.pow(math.sin(k), 2.0))) / l) * (k / t_1))
	else:
		tmp = 2.0 / ((t_2 / l) * (t_2 / t_1))
	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(l * cos(k))
	t_2 = Float64(k * Float64(sin(k) * sqrt(t)))
	tmp = 0.0
	if (t <= 3.4056655695663573e-273)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * (sin(k) ^ 2.0))) / l) * Float64(k / t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 / l) * Float64(t_2 / t_1)));
	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 = l * cos(k);
	t_2 = k * (sin(k) * sqrt(t));
	tmp = 0.0;
	if (t <= 3.4056655695663573e-273)
		tmp = 2.0 / (((k * (t * (sin(k) ^ 2.0))) / l) * (k / t_1));
	else
		tmp = 2.0 / ((t_2 / l) * (t_2 / t_1));
	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[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 3.4056655695663573e-273], N[(2.0 / N[(N[(N[(k * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 / l), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 3.40566556956635729e-273

   1. Initial program 48.8

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

   2. Simplified 41.2

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

   3. Taylor expanded in t around 0 22.6

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

   4. Applied egg-rr 21.9

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

   5. Applied egg-rr 18.4

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

   6. Applied egg-rr 8.8

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

   if 3.40566556956635729e-273 < t

   1. Initial program 47.6

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

   2. Simplified 39.8

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

   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 21.2

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

   5. Applied egg-rr 18.1

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

   6. Applied egg-rr 3.5

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

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4056655695663573 \cdot 10^{-273}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell} \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell} \cdot \frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\ell \cdot \cos k}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))