Toniolo and Linder, Equation (2)

Average Error: 10.3 → 0.9

Time: 27.2s

Precision: binary64

Cost: 27336

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot -1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+38}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 2\right) + 2\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ \end{array} \]

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) -4e+124)
   (asin (* (/ (* l (sqrt 0.5)) t) -1.0))
   (if (<= (/ t l) 1e+38)
     (asin
      (sqrt
       (/
        (- 1.0 (+ (- (pow (/ Om Omc) 2.0) 2.0) 2.0))
        (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* (/ (* (sqrt 0.5) l) t) 1.0)))))

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) <= -4e+124) {
		tmp = asin((((l * sqrt(0.5)) / t) * -1.0));
	} else if ((t / l) <= 1e+38) {
		tmp = asin(sqrt(((1.0 - ((pow((Om / Omc), 2.0) - 2.0) + 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((((sqrt(0.5) * l) / t) * 1.0));
	}
	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) <= (-4d+124)) then
        tmp = asin((((l * sqrt(0.5d0)) / t) * (-1.0d0)))
    else if ((t / l) <= 1d+38) then
        tmp = asin(sqrt(((1.0d0 - ((((om / omc) ** 2.0d0) - 2.0d0) + 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((((sqrt(0.5d0) * l) / t) * 1.0d0))
    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) <= -4e+124) {
		tmp = Math.asin((((l * Math.sqrt(0.5)) / t) * -1.0));
	} else if ((t / l) <= 1e+38) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Math.pow((Om / Omc), 2.0) - 2.0) + 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((((Math.sqrt(0.5) * l) / t) * 1.0));
	}
	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) <= -4e+124:
		tmp = math.asin((((l * math.sqrt(0.5)) / t) * -1.0))
	elif (t / l) <= 1e+38:
		tmp = math.asin(math.sqrt(((1.0 - ((math.pow((Om / Omc), 2.0) - 2.0) + 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((((math.sqrt(0.5) * l) / t) * 1.0))
	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) <= -4e+124)
		tmp = asin(Float64(Float64(Float64(l * sqrt(0.5)) / t) * -1.0));
	elseif (Float64(t / l) <= 1e+38)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64((Float64(Om / Omc) ^ 2.0) - 2.0) + 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l) / t) * 1.0));
	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) <= -4e+124)
		tmp = asin((((l * sqrt(0.5)) / t) * -1.0));
	elseif ((t / l) <= 1e+38)
		tmp = asin(sqrt(((1.0 - ((((Om / Omc) ^ 2.0) - 2.0) + 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((((sqrt(0.5) * l) / t) * 1.0));
	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], -4e+124], N[ArcSin[N[(N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+38], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] - 2.0), $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], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] * 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999979e124

   1. Initial program 30.9

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

   2. Taylor expanded in t around -inf 8.0

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

   3. Simplified 8.0

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

      [Start] 8.0	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      rational.json-simplify-43 [=>] 8.0	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot -1\right)\right)} \]
      rational.json-simplify-2 [=>] 8.0	\[ \sin^{-1} \left(\frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t} \cdot \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot -1\right)\right) \]
      rational.json-simplify-9 [=>] 8.0	\[ \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \color{blue}{\left(-\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]

   4. Taylor expanded in Om around 0 0.7

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

   if -3.99999999999999979e124 < (/.f64 t l) < 9.99999999999999977e37

   1. Initial program 0.9

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

   2. Applied egg-rr 0.9

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\left(\left(0 - \left(-1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right) - 1\right)}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   3. Applied egg-rr 0.9

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

   if 9.99999999999999977e37 < (/.f64 t l)

   1. Initial program 21.7

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

   2. Taylor expanded in t around inf 8.2

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

   3. Taylor expanded in Om around 0 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot -1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+38}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 2\right) + 2\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.0
Cost	27096

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ t_2 := \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2.35 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{-258}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot -1\right)\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.22 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 2
Error	0.9
Cost	27080

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot -1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+38}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ \end{array} \]

Alternative 3
Error	23.8
Cost	13776

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{-258}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot -1\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 4
Error	23.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.45 \cdot 10^{-90}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 5
Error	31.8
Cost	6464

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

Reproduce

herbie shell --seed 2023077 
(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)))))))