Toniolo and Linder, Equation (2)

Average Error: 10.4 → 0.9

Time: 12.0s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\frac{1}{t} \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

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+56)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 1e+27)
     (asin
      (sqrt
       (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* l (* (/ 1.0 t) (sqrt 0.5)))))))

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+56) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+27) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((l * ((1.0 / t) * sqrt(0.5))));
	}
	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+56)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 1d+27) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((l * ((1.0d0 / t) * sqrt(0.5d0))))
    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+56) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+27) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((l * ((1.0 / t) * Math.sqrt(0.5))));
	}
	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+56:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 1e+27:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((l * ((1.0 / t) * math.sqrt(0.5))))
	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+56)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 1e+27)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(l * Float64(Float64(1.0 / t) * sqrt(0.5))));
	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+56)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 1e+27)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((l * ((1.0 / t) * sqrt(0.5))));
	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+56], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+27], 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], N[ArcSin[N[(l * N[(N[(1.0 / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.00000000000000024e56

   1. Initial program 23.9

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

   2. Taylor expanded in Om around 0 35.8

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

   3. Taylor expanded in t around -inf 0.7

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

   4. Simplified 0.7

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

      [Start] 0.7	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      rational.json-simplify-2 [=>] 0.7	\[ \sin^{-1} \left(-1 \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]
      rational.json-simplify-49 [=>] 0.7	\[ \sin^{-1} \left(-1 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)}\right) \]
      rational.json-simplify-43 [=>] 0.7	\[ \sin^{-1} \color{blue}{\left(\sqrt{0.5} \cdot \left(\frac{\ell}{t} \cdot -1\right)\right)} \]
      rational.json-simplify-9 [=>] 0.7	\[ \sin^{-1} \left(\sqrt{0.5} \cdot \color{blue}{\left(-\frac{\ell}{t}\right)}\right) \]

   if -5.00000000000000024e56 < (/.f64 t l) < 1e27

   1. Initial program 1.0

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

   if 1e27 < (/.f64 t l)

   1. Initial program 20.7

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

   2. Taylor expanded in Om around 0 34.2

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

   3. Taylor expanded in t around inf 0.8

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      rational.json-simplify-49 [=>] 0.8	\[ \sin^{-1} \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)} \]

   5. Applied egg-rr 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\frac{1}{t} \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\frac{1}{t} \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

Alternative 2
Error	13.3
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+219}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 3
Error	2.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	2.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	23.3
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	32.3
Cost	6464

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

Reproduce

herbie shell --seed 2023064 
(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)))))))