Toniolo and Linder, Equation (2)

Average Error: 10.3 → 1.0

Time: 13.2s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]

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) -2e+157)
   (asin (- (/ (* l (sqrt 0.5)) t)))
   (if (<= (/ t l) 1e+154)
     (asin
      (sqrt
       (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (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) <= -2e+157) {
		tmp = asin(-((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} 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) <= (-2d+157)) then
        tmp = asin(-((l * sqrt(0.5d0)) / t))
    else if ((t / l) <= 1d+154) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    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) <= -2e+157) {
		tmp = Math.asin(-((l * Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 1e+154) {
		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(((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) <= -2e+157:
		tmp = math.asin(-((l * math.sqrt(0.5)) / t))
	elif (t / l) <= 1e+154:
		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(((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) <= -2e+157)
		tmp = asin(Float64(-Float64(Float64(l * sqrt(0.5)) / t)));
	elseif (Float64(t / l) <= 1e+154)
		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(Float64(sqrt(0.5) * 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) <= -2e+157)
		tmp = asin(-((l * sqrt(0.5)) / t));
	elseif ((t / l) <= 1e+154)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	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], -2e+157], N[ArcSin[(-N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], 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[(N[(N[Sqrt[0.5], $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.99999999999999997e157

   1. Initial program 33.7

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

   2. Taylor expanded in Om around 0 33.7

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

   3. Taylor expanded in t around -inf 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      rational_best_45_simplify-18 [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot -1\right)} \]
      rational_best_45_simplify-63 [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
      rational_best_45_simplify-18 [=>] 0.6	\[ \sin^{-1} \left(-\frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

   if -1.99999999999999997e157 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 1.1

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 34.5

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

   2. Taylor expanded in Om around 0 34.5

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

   3. Taylor expanded in t around inf 0.7

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	16.4
Cost	26832

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right)\\ t_2 := \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-160}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	23.5
Cost	13712

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ t_2 := \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+298}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	23.7
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	31.6
Cost	6464

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

Reproduce

herbie shell --seed 2023102 
(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)))))))