Toniolo and Linder, Equation (2)

Average Error: 10.1 → 0.8

Time: 12.0s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

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+131)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 5e+125)
     (asin
      (sqrt
       (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* l (/ (sqrt 0.5) 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+131) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 5e+125) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((l * (sqrt(0.5) / 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+131)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 5d+125) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((l * (sqrt(0.5d0) / 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+131) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 5e+125) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / 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+131:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 5e+125:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / 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+131)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 5e+125)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / 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+131)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 5e+125)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((l * (sqrt(0.5) / 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+131], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+125], 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], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.99999999999999995e131

   1. Initial program 29.8

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

   2. Taylor expanded in Om around 0 33.5

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

   3. Simplified 33.5

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 4 + 2}}\right)} \]
      Proof

      [Start] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      rational.json-simplify-35 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 + 1}{\left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}}}\right) \]
      metadata-eval [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{\color{blue}{2}}{\left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}}\right) \]
      rational.json-simplify-41 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{1 + \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)}}}\right) \]
      rational.json-simplify-17 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) - -1}}}\right) \]
      rational.json-simplify-41 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(1 + \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)} - -1}}\right) \]
      rational.json-simplify-48 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 - -1\right)}}}\right) \]
      rational.json-simplify-2 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\left(\color{blue}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 2} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 - -1\right)}}\right) \]
      rational.json-simplify-51 [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\frac{{t}^{2}}{{\ell}^{2}} \cdot \left(2 + 2\right)} + \left(1 - -1\right)}}\right) \]
      metadata-eval [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot \color{blue}{4} + \left(1 - -1\right)}}\right) \]
      metadata-eval [=>] 33.5	\[ \sin^{-1} \left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 4 + \color{blue}{2}}}\right) \]

   4. Taylor expanded in t around inf 33.3

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{0.5 \cdot \frac{{\ell}^{2}}{{t}^{2}}}}\right) \]

   5. Taylor expanded in l around -inf 0.6

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

   6. Simplified 0.6

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

      [Start] 0.6	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      rational.json-simplify-2 [=>] 0.6	\[ \sin^{-1} \left(-1 \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]
      rational.json-simplify-49 [=>] 0.6	\[ \sin^{-1} \left(-1 \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)}\right) \]
      rational.json-simplify-43 [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(\sqrt{0.5} \cdot \left(\frac{\ell}{t} \cdot -1\right)\right)} \]
      rational.json-simplify-9 [=>] 0.6	\[ \sin^{-1} \left(\sqrt{0.5} \cdot \color{blue}{\left(-\frac{\ell}{t}\right)}\right) \]

   if -4.99999999999999995e131 < (/.f64 t l) < 4.99999999999999962e125

   1. Initial program 0.9

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

   if 4.99999999999999962e125 < (/.f64 t l)

   1. Initial program 30.0

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

   2. Taylor expanded in Om around 0 34.0

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

   3. Simplified 34.0

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 4 + 2}}\right)} \]
      Proof

      [Start] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      rational.json-simplify-35 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 + 1}{\left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}}}\right) \]
      metadata-eval [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{\color{blue}{2}}{\left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}}\right) \]
      rational.json-simplify-41 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{1 + \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)}}}\right) \]
      rational.json-simplify-17 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) - -1}}}\right) \]
      rational.json-simplify-41 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(1 + \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)} - -1}}\right) \]
      rational.json-simplify-48 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 - -1\right)}}}\right) \]
      rational.json-simplify-2 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\left(\color{blue}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 2} + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) + \left(1 - -1\right)}}\right) \]
      rational.json-simplify-51 [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\color{blue}{\frac{{t}^{2}}{{\ell}^{2}} \cdot \left(2 + 2\right)} + \left(1 - -1\right)}}\right) \]
      metadata-eval [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot \color{blue}{4} + \left(1 - -1\right)}}\right) \]
      metadata-eval [=>] 34.0	\[ \sin^{-1} \left(\sqrt{\frac{2}{\frac{{t}^{2}}{{\ell}^{2}} \cdot 4 + \color{blue}{2}}}\right) \]

   4. Taylor expanded in t around inf 33.5

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{0.5 \cdot \frac{{\ell}^{2}}{{t}^{2}}}}\right) \]

   5. Taylor expanded in l around 0 0.6

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

   6. Simplified 0.6

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

      [Start] 0.6	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      rational.json-simplify-49 [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 2
Error	2.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 3
Error	22.9
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{+41}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.9
Cost	13384

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+42}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 5
Error	31.8
Cost	6464

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

Reproduce

herbie shell --seed 2023068 
(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)))))))