Toniolo and Linder, Equation (2)

Average Error: 10.5 → 1.4

Time: 14.7s

Precision: binary64

Cost: 20484

Math

\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}} \cdot \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))

↓

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+155)
   (asin
    (* (sqrt (- 1.0 (* Om (/ Om (* Omc Omc))))) (* l (- (/ (sqrt 0.5) t)))))
   (if (<= (/ t l) 1.1e+125)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
}

↓

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+155) {
		tmp = asin((sqrt((1.0 - (Om * (Om / (Omc * Omc))))) * (l * -(sqrt(0.5) / t))));
	} else if ((t / l) <= 1.1e+125) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
end function

↓

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-5d+155)) then
        tmp = asin((sqrt((1.0d0 - (om * (om / (omc * omc))))) * (l * -(sqrt(0.5d0) / t))))
    else if ((t / l) <= 1.1d+125) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
}

↓

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+155) {
		tmp = Math.asin((Math.sqrt((1.0 - (Om * (Om / (Omc * Omc))))) * (l * -(Math.sqrt(0.5) / t))));
	} else if ((t / l) <= 1.1e+125) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))

↓

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+155:
		tmp = math.asin((math.sqrt((1.0 - (Om * (Om / (Omc * Omc))))) * (l * -(math.sqrt(0.5) / t))))
	elif (t / l) <= 1.1e+125:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

function code(t, l, Om, Omc)
	return asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))))
end

↓

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+155)
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc))))) * Float64(l * Float64(-Float64(sqrt(0.5) / t)))));
	elseif (Float64(t / l) <= 1.1e+125)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
end

↓

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+155)
		tmp = asin((sqrt((1.0 - (Om * (Om / (Omc * Omc))))) * (l * -(sqrt(0.5) / t))));
	elseif ((t / l) <= 1.1e+125)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+155], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1.1e+125], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999999e155

   1. Initial program 35.5

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Applied egg-rr 35.5

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   3. Applied egg-rr 35.5

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around -inf 7.7

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]

   5. Simplified 3.9

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om}{Omc \cdot Omc} \cdot Om} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om (*.f64 Omc Omc)) Om))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))) Om))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= associate-/r/_binary64 (/.f64 Om (/.f64 (pow.f64 Omc 2) Om))))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 Om Om) (pow.f64 Omc 2))))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 15 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (pow.f64 Omc 2)))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 29 points increase in error, 20 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error

   if -4.9999999999999999e155 < (/.f64 t l) < 1.09999999999999995e125

   1. Initial program 1.0

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Applied egg-rr 1.0

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   3. Taylor expanded in t around 0 16.8

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]

   4. Simplified 1.0

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      Proof
      (*.f64 (/.f64 t l) (/.f64 t l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t t) (*.f64 l l))): 81 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error

   if 1.09999999999999995e125 < (/.f64 t l)

   1. Initial program 30.7

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around inf 7.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

   3. Taylor expanded in Om around 0 0.7

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   4. Simplified 0.7

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)} \]
      Proof
      (*.f64 (/.f64 l t) (sqrt.f64 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 t (sqrt.f64 1/2)))): 36 points increase in error, 31 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l (sqrt.f64 1/2)) t)): 27 points increase in error, 27 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) l)) t): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}} \cdot \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	32832

\[\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

Alternative 2
Error	1.0
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 3
Error	2.0
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-10}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 4
Error	24.6
Cost	13648

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\ell \leq -4.24846661087336 \cdot 10^{-74}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.555975014374001 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.141061690224584 \cdot 10^{-188}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.4084194755947404 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 5
Error	24.6
Cost	13648

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.24846661087336 \cdot 10^{-74}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.555975014374001 \cdot 10^{-215}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\ell \leq 3.141061690224584 \cdot 10^{-188}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.4084194755947404 \cdot 10^{-26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 6
Error	13.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+212}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-10}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7
Error	2.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-10}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 8
Error	32.0
Cost	6464

\[\sin^{-1} 1 \]

Reproduce

herbie shell --seed 2022306 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))