Toniolo and Linder, Equation (2)

Average Error: 10.4 → 1.1

Time: 18.5s

Precision: binary64

Cost: 32832

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 10.4

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

2. Applied egg-rr 1.1

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

3. Simplified 1.1

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
   Proof
   (/.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) 1)) (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (/.f64 1 (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))))): 0 points increase in error, 0 points decrease in error

4. Final simplification 1.1

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

Alternatives

Alternative 1
Error	1.6
Cost	19712

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

Alternative 2
Error	12.7
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	12.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	32.0
Cost	7104

\[\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right) \]

Alternative 8
Error	32.2
Cost	6464

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

Reproduce

herbie shell --seed 2022321 
(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)))))))