Toniolo and Linder, Equation (2)

Average Error: 10.0 → 1.0

Time: 34.9s

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 10.0

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

2. Applied egg-rr 1.0

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

3. Simplified 1.0

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
   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 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))): 26 points increase in error, 0 points decrease in error
   (/.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 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 (/.f64 (pow.f64 Om 2) (pow.f64 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 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 1 (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite=> unpow2_binary64 (*.f64 Om Om)) (pow.f64 Omc 2)))) (/.f64 1 (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (*.f64 Om Om) (Rewrite=> unpow2_binary64 (*.f64 Omc Omc))))) (/.f64 1 (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (/.f64 1 (hypot.f64 1 (*.f64 (/.f64 t l) (sqrt.f64 2))))): 0 points increase in error, 26 points decrease in error
   (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= unpow2_binary64 (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 1.0

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

Alternatives

Alternative 1
Error	1.6
Cost	19712

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

Alternative 2
Error	11.6
Cost	14144

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

Alternative 3
Error	9.9
Cost	14144

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

Alternative 4
Error	16.0
Cost	14084

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{\left(0.5 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}}{t}\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.4
Cost	13896

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	31.6
Cost	13376

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

Alternative 7
Error	31.8
Cost	7104

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

Alternative 8
Error	31.9
Cost	6464

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

Reproduce

herbie shell --seed 2022325 
(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)))))))