Toniolo and Linder, Equation (2)

Average Error: 9.9 → 1.0

Time: 13.1s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

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) -4e+32)
   (asin (- (/ (sqrt 0.5) (/ t l))))
   (if (<= (/ t l) 2e+108)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (/ l (/ 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) <= -4e+32) {
		tmp = asin(-(sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 2e+108) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((l / (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) <= (-4d+32)) then
        tmp = asin(-(sqrt(0.5d0) / (t / l)))
    else if ((t / l) <= 2d+108) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((l / (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) <= -4e+32) {
		tmp = Math.asin(-(Math.sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 2e+108) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((l / (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) <= -4e+32:
		tmp = math.asin(-(math.sqrt(0.5) / (t / l)))
	elif (t / l) <= 2e+108:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((l / (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) <= -4e+32)
		tmp = asin(Float64(-Float64(sqrt(0.5) / Float64(t / l))));
	elseif (Float64(t / l) <= 2e+108)
		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(l / Float64(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) <= -4e+32)
		tmp = asin(-(sqrt(0.5) / (t / l)));
	elseif ((t / l) <= 2e+108)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((l / (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], -4e+32], N[ArcSin[(-N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+108], 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[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.00000000000000021e32

   1. Initial program 21.3

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

   2. Applied egg-rr 21.3

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

   3. Taylor expanded in Om around 0 35.2

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

   4. Simplified 35.2

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

   5. Taylor expanded in t around -inf 0.8

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

   6. Simplified 1.4

      \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
      Proof
      (neg.f64 (/.f64 (sqrt.f64 1/2) (/.f64 t l))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 34 points increase in error, 31 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 -4.00000000000000021e32 < (/.f64 t l) < 2.0000000000000001e108

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 1.0

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

   if 2.0000000000000001e108 < (/.f64 t l)

   1. Initial program 27.8

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

   2. Applied egg-rr 27.8

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

   3. Taylor expanded in Om around 0 35.4

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

   4. Simplified 35.4

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

   5. Taylor expanded in t around inf 0.7

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

   6. Simplified 0.7

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)} \]
      Proof
      (/.f64 l (/.f64 t (sqrt.f64 1/2))): 0 points increase in error, 0 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	32832

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

Alternative 2
Error	1.7
Cost	19712

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

Alternative 3
Error	1.5
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 5
Error	13.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -20:\\ \;\;\;\;\sin^{-1} \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 7
Error	23.3
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+79}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	23.2
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	31.2
Cost	6464

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

Reproduce

herbie shell --seed 2022338 
(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)))))))