Toniolo and Linder, Equation (2)

Average Error: 10.5 → 1.1

Time: 12.5s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \]

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
 (let* ((t_1 (/ t (sqrt 0.5))))
   (if (<= (/ t l) -5e+161)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 2e+107)
       (asin
        (sqrt
         (/
          (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
          (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (/ l t_1))))))

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 t_1 = t / sqrt(0.5);
	double tmp;
	if ((t / l) <= -5e+161) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 2e+107) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((l / t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t / sqrt(0.5d0)
    if ((t / l) <= (-5d+161)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 2d+107) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((l / t_1))
    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 t_1 = t / Math.sqrt(0.5);
	double tmp;
	if ((t / l) <= -5e+161) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 2e+107) {
		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_1));
	}
	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):
	t_1 = t / math.sqrt(0.5)
	tmp = 0
	if (t / l) <= -5e+161:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 2e+107:
		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_1))
	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)
	t_1 = Float64(t / sqrt(0.5))
	tmp = 0.0
	if (Float64(t / l) <= -5e+161)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 2e+107)
		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 / t_1));
	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)
	t_1 = t / sqrt(0.5);
	tmp = 0.0;
	if ((t / l) <= -5e+161)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 2e+107)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((l / t_1));
	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_] := Block[{t$95$1 = N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+161], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+107], 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 / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999997e161

   1. Initial program 33.6

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

   2. Applied egg-rr 33.6

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

   3. Taylor expanded in Om around 0 33.6

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

   4. Simplified 33.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
      Proof
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (*.f64 t (/.f64 t (*.f64 l l))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (*.f64 t (/.f64 t (Rewrite<= unpow2_binary64 (pow.f64 l 2)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 t t) (pow.f64 l 2))))))): 46 points increase in error, 1 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in t around -inf 0.5

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

   6. Simplified 0.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)} \]
      Proof
      (/.f64 (neg.f64 l) (/.f64 t (sqrt.f64 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 l (/.f64 t (sqrt.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l (sqrt.f64 1/2)) t))): 34 points increase in error, 38 points decrease in error
      (neg.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.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 -4.9999999999999997e161 < (/.f64 t l) < 1.9999999999999999e107

   1. Initial program 1.3

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

   2. Applied egg-rr 1.3

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

   3. Applied egg-rr 1.3

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

   if 1.9999999999999999e107 < (/.f64 t l)

   1. Initial program 29.1

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

   2. Applied egg-rr 29.1

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

   3. Taylor expanded in Om around 0 35.7

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

   4. Simplified 32.8

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
      Proof
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (*.f64 t (/.f64 t (*.f64 l l))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (*.f64 t (/.f64 t (Rewrite<= unpow2_binary64 (pow.f64 l 2)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 t t) (pow.f64 l 2))))))): 46 points increase in error, 1 points decrease in error
      (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in t around inf 0.5

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

   6. Simplified 0.5

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)} \]
      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)): 34 points increase in error, 38 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 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

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	1.6
Cost	19712

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

Alternative 3
Error	1.6
Cost	14152

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+46}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t}{\frac{\ell}{\frac{t}{\ell}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \]

Alternative 4
Error	1.6
Cost	14152

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \]

Alternative 5
Error	2.1
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.9
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	13.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+221}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Error	2.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Error	23.7
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.46 \cdot 10^{-61}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10
Error	32.2
Cost	6464

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

Reproduce

herbie shell --seed 2022331 
(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)))))))