Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 14.3

Time: 14.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)\\ t_3 := {\sin k}^{2}\\ t_4 := \ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left({\left(\sqrt[3]{\sin k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot t_3\right)\right)}\right)\\ \mathbf{if}\;t_2 \leq -70969323.52844514:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 14593031620471792:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{\tan k \cdot \left(2 + t_1\right)}{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k}}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{t_3}\right)\\ \end{array} \]

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
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        (t_3 (pow (sin k) 2.0))
        (t_4
         (*
          l
          (*
           2.0
           (/
            l
            (fma
             2.0
             (pow (* (pow (cbrt (sin k)) 2.0) (/ t (cbrt (cos k)))) 3.0)
             (* (/ (* k k) (cos k)) (* t t_3))))))))
   (if (<= t_2 -70969323.52844514)
     t_4
     (if (<= t_2 14593031620471792.0)
       (*
        l
        (/
         1.0
         (/ (* (tan k) (+ 2.0 t_1)) (* l (/ (/ 2.0 (pow t 3.0)) (sin k))))))
       (if (<= t_2 INFINITY)
         t_4
         (* l (* (/ 2.0 (* k (* t k))) (/ (* l (cos k)) t_3))))))))

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((k / t), 2.0);
	double t_2 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1));
	double t_3 = pow(sin(k), 2.0);
	double t_4 = l * (2.0 * (l / fma(2.0, pow((pow(cbrt(sin(k)), 2.0) * (t / cbrt(cos(k)))), 3.0), (((k * k) / cos(k)) * (t * t_3)))));
	double tmp;
	if (t_2 <= -70969323.52844514) {
		tmp = t_4;
	} else if (t_2 <= 14593031620471792.0) {
		tmp = l * (1.0 / ((tan(k) * (2.0 + t_1)) / (l * ((2.0 / pow(t, 3.0)) / sin(k)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = l * ((2.0 / (k * (t * k))) * ((l * cos(k)) / t_3));
	}
	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(k / t) ^ 2.0
	t_2 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))
	t_3 = sin(k) ^ 2.0
	t_4 = Float64(l * Float64(2.0 * Float64(l / fma(2.0, (Float64((cbrt(sin(k)) ^ 2.0) * Float64(t / cbrt(cos(k)))) ^ 3.0), Float64(Float64(Float64(k * k) / cos(k)) * Float64(t * t_3))))))
	tmp = 0.0
	if (t_2 <= -70969323.52844514)
		tmp = t_4;
	elseif (t_2 <= 14593031620471792.0)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(tan(k) * Float64(2.0 + t_1)) / Float64(l * Float64(Float64(2.0 / (t ^ 3.0)) / sin(k))))));
	elseif (t_2 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * cos(k)) / t_3)));
	end
	return 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[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = 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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(l * N[(2.0 * N[(l / N[(2.0 * N[Power[N[(N[Power[N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Power[N[Cos[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -70969323.52844514], t$95$4, If[LessEqual[t$95$2, 14593031620471792.0], N[(l * N[(1.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$4, N[(l * N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < -70969323.5284451395 or 14593031620471792 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 13.7

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

   2. Simplified 12.1

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

   3. Taylor expanded in l around inf 17.5

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

   4. Simplified 17.5

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

   5. Applied egg-rr 16.3

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

   6. Applied egg-rr 6.5

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

   if -70969323.5284451395 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 14593031620471792

   1. Initial program 27.8

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

   2. Simplified 21.4

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

   3. Applied egg-rr 14.7

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   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 59.0

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

   3. Taylor expanded in t around 0 34.9

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

   4. Simplified 26.4

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

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq -70969323.52844514:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left({\left(\sqrt[3]{\sin k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq 14593031620471792:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k}}}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left({\left(\sqrt[3]{\sin k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))))