Toniolo and Linder, Equation (2)

Average Error: 10.1 → 1.0

Time: 12.9s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

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+158)
   (asin (/ (* l (- (sqrt 0.5))) t))
   (if (<= (/ t l) 5e+58)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (* (sqrt 0.5) (/ l t))))))

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+158) {
		tmp = asin(((l * -sqrt(0.5)) / t));
	} else if ((t / l) <= 5e+58) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	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+158)) then
        tmp = asin(((l * -sqrt(0.5d0)) / t))
    else if ((t / l) <= 5d+58) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    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+158) {
		tmp = Math.asin(((l * -Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 5e+58) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	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+158:
		tmp = math.asin(((l * -math.sqrt(0.5)) / t))
	elif (t / l) <= 5e+58:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	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+158)
		tmp = asin(Float64(Float64(l * Float64(-sqrt(0.5))) / t));
	elseif (Float64(t / l) <= 5e+58)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	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+158)
		tmp = asin(((l * -sqrt(0.5)) / t));
	elseif ((t / l) <= 5e+58)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	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+158], N[ArcSin[N[(N[(l * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+58], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999996e158

   1. Initial program 34.1

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

   2. Applied egg-rr 1.2

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

   3. Taylor expanded in Om around 0 34.1

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

   4. Simplified 34.1

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \left(t \cdot t\right)}}\right)} \]
      Proof
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 2 (*.f64 l l)) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 2) (sqrt.f64 2))) (*.f64 l l)) (*.f64 t t))))): 17 points increase in error, 1 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 2) 2)) (*.f64 l l)) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (pow.f64 (sqrt.f64 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (pow.f64 (sqrt.f64 2) 2) (pow.f64 l 2)) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (sqrt.f64 2) 2) (pow.f64 t 2)) (pow.f64 l 2)))))): 1 points increase in error, 2 points decrease in error

   5. Taylor expanded in t around -inf 0.6

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

   6. Simplified 0.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)} \]
      Proof
      (/.f64 (*.f64 l (neg.f64 (sqrt.f64 1/2))) t): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 l (sqrt.f64 1/2)))) t): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) l))) t): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 0 points increase in error, 0 points decrease in error

   if -4.9999999999999996e158 < (/.f64 t l) < 4.99999999999999986e58

   1. Initial program 1.1

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.1

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

   if 4.99999999999999986e58 < (/.f64 t l)

   1. Initial program 23.4

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

   2. Applied egg-rr 1.0

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

   3. Taylor expanded in Om around 0 34.0

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

   4. Simplified 34.1

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \left(t \cdot t\right)}}\right)} \]
      Proof
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 2 (*.f64 l l)) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 2) (sqrt.f64 2))) (*.f64 l l)) (*.f64 t t))))): 17 points increase in error, 1 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 2) 2)) (*.f64 l l)) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (pow.f64 (sqrt.f64 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 (/.f64 (pow.f64 (sqrt.f64 2) 2) (pow.f64 l 2)) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (pow.f64 (sqrt.f64 2) 2) (pow.f64 t 2)) (pow.f64 l 2)))))): 1 points increase in error, 2 points decrease in error

   5. Taylor expanded in l around 0 0.9

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

   6. Simplified 1.0

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)} \]
      Proof
      (*.f64 (/.f64 l t) (sqrt.f64 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 t (sqrt.f64 1/2)))): 38 points increase in error, 25 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l (sqrt.f64 1/2)) t)): 30 points increase in error, 32 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 1/2) l)) t): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26624

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

Alternative 2
Error	1.6
Cost	19712

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

Alternative 3
Error	0.9
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 5
Error	1.6
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 6
Error	13.3
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{t}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 8
Error	31.5
Cost	6464

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

Reproduce

herbie shell --seed 2022300 
(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)))))))