Toniolo and Linder, Equation (2)

Average Error: 10.1 → 1.0

Time: 20.4s

Precision: binary64

Cost: 26432

Math

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

↓

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

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
 (asin
  (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (hypot 1.0 (* (/ t l) (sqrt 2.0))))))

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

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) {
	return Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) / Math.hypot(1.0, ((t / l) * Math.sqrt(2.0)))));
}

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

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)
	return asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / hypot(1.0, Float64(Float64(t / l) * sqrt(2.0)))))
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 = code(t, l, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) / hypot(1.0, ((t / l) * sqrt(2.0)))));
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_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 10.1

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

2. Applied egg-rr 1.0

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

3. Final simplification 1.0

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

Alternatives

Alternative 1
Error	13.3
Cost	20428

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ t_2 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\ \mathbf{if}\;\ell \leq -5.469267190637758 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -6.088768676395897 \cdot 10^{-214}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_2 \cdot \left(\ell \cdot \sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\ell \leq 2.1586936958426744 \cdot 10^{-224}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.0
Cost	20224

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

Alternative 3
Error	17.5
Cost	14484

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ t_2 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\frac{t \cdot \left(2 \cdot t\right)}{\ell}}{\ell}}}\right)\\ t_3 := \sin^{-1} \left(\sqrt{1 + \left(1 + \left(-1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)\right)}\right)\\ t_4 := t \cdot \sqrt{2}\\ \mathbf{if}\;\ell \leq -1.621534069950617 \cdot 10^{+164}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -5.469267190637758 \cdot 10^{-71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -6.088768676395897 \cdot 10^{-214}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell}{t_4}\right)\\ \mathbf{elif}\;\ell \leq -3.244181907018592 \cdot 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.1586936958426744 \cdot 10^{-224}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_4} \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 4.474945657304066 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	23.9
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{if}\;\ell \leq -0.09152764601887171:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.145134457015355 \cdot 10^{-30}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	31.3
Cost	64

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

Reproduce

herbie shell --seed 2022289 
(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)))))))