Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 82.8%

Time: 23.6s

Alternatives: 11

Speedup: 75.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 33.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := 2 \cdot {t}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{t_3 + t_3}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(t_2 + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{t_3}{x} + \frac{t_3}{{x}^{3}}\right)}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0))))
        (t_2 (* 2.0 (pow t 2.0)))
        (t_3 (+ t_2 (pow l 2.0))))
   (if (<= t -5.5e+19)
     (- t_1)
     (if (<= t -2.85e-213)
       (/
        t
        (/
         (sqrt
          (+
           (+
            (/ (+ t_3 t_3) (pow x 2.0))
            (+
             (* 2.0 (/ (pow t 2.0) x))
             (+
              (* 2.0 (/ (pow t 2.0) (pow x 3.0)))
              (+ t_2 (+ (/ (pow l 2.0) x) (/ (pow l 2.0) (pow x 3.0)))))))
           (+ (/ t_3 x) (/ t_3 (pow x 3.0)))))
         (sqrt 2.0)))
       (if (<= t 8.6e-206)
         (/ (* t (sqrt (fma x 0.5 -0.5))) (/ l (sqrt 2.0)))
         t_1)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double t_2 = 2.0 * pow(t, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double tmp;
	if (t <= -5.5e+19) {
		tmp = -t_1;
	} else if (t <= -2.85e-213) {
		tmp = t / (sqrt(((((t_3 + t_3) / pow(x, 2.0)) + ((2.0 * (pow(t, 2.0) / x)) + ((2.0 * (pow(t, 2.0) / pow(x, 3.0))) + (t_2 + ((pow(l, 2.0) / x) + (pow(l, 2.0) / pow(x, 3.0))))))) + ((t_3 / x) + (t_3 / pow(x, 3.0))))) / sqrt(2.0));
	} else if (t <= 8.6e-206) {
		tmp = (t * sqrt(fma(x, 0.5, -0.5))) / (l / sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	t_2 = Float64(2.0 * (t ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	tmp = 0.0
	if (t <= -5.5e+19)
		tmp = Float64(-t_1);
	elseif (t <= -2.85e-213)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(Float64(2.0 * Float64((t ^ 2.0) / (x ^ 3.0))) + Float64(t_2 + Float64(Float64((l ^ 2.0) / x) + Float64((l ^ 2.0) / (x ^ 3.0))))))) + Float64(Float64(t_3 / x) + Float64(t_3 / (x ^ 3.0))))) / sqrt(2.0)));
	elseif (t <= 8.6e-206)
		tmp = Float64(Float64(t * sqrt(fma(x, 0.5, -0.5))) / Float64(l / sqrt(2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.5e+19], (-t$95$1), If[LessEqual[t, -2.85e-213], N[(t / N[(N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / x), $MachinePrecision] + N[(t$95$3 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e-206], N[(N[(t * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.5e19

   1. Initial program 45.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 97.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.4%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 97.4%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 97.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -5.5e19 < t < -2.84999999999999997e-213

   1. Initial program 42.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around -inf 79.6%

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right) - \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) - \left(-1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{{x}^{3}}\right)}}}{\sqrt{2}}} \]

   if -2.84999999999999997e-213 < t < 8.6000000000000005e-206

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 7.3%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.3%

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

      2. associate--l+ 35.4%

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

      3. sub-neg 35.4%

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

      4. metadata-eval 35.4%

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

      5. +-commutative 35.4%

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

      6. sub-neg 35.4%

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

      7. metadata-eval 35.4%

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

      8. +-commutative 35.4%

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

      9. associate-/l* 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in x around 0 49.7%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.6%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot x - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

      2. *-commutative 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot 0.5} - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      3. fma-neg 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      4. metadata-eval 49.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

   9. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

   if 8.6000000000000005e-206 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{3}}\right)\right)\right)\right)\right) + \left(\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{{x}^{3}}\right)}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 2: 82.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := 2 \cdot {t}^{2}\\ t_3 := t_2 + {\ell}^{2}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+18}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{t_3}{x} + \left(\frac{t_3 + t_3}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(t_2 + \frac{{\ell}^{2}}{x}\right)\right)\right)}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0))))
        (t_2 (* 2.0 (pow t 2.0)))
        (t_3 (+ t_2 (pow l 2.0))))
   (if (<= t -6.8e+18)
     (- t_1)
     (if (<= t -2.85e-213)
       (/
        t
        (/
         (sqrt
          (+
           (/ t_3 x)
           (+
            (/ (+ t_3 t_3) (pow x 2.0))
            (+ (* 2.0 (/ (pow t 2.0) x)) (+ t_2 (/ (pow l 2.0) x))))))
         (sqrt 2.0)))
       (if (<= t 8e-206)
         (/ (* t (sqrt (fma x 0.5 -0.5))) (/ l (sqrt 2.0)))
         t_1)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double t_2 = 2.0 * pow(t, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double tmp;
	if (t <= -6.8e+18) {
		tmp = -t_1;
	} else if (t <= -2.85e-213) {
		tmp = t / (sqrt(((t_3 / x) + (((t_3 + t_3) / pow(x, 2.0)) + ((2.0 * (pow(t, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x)))))) / sqrt(2.0));
	} else if (t <= 8e-206) {
		tmp = (t * sqrt(fma(x, 0.5, -0.5))) / (l / sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	t_2 = Float64(2.0 * (t ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	tmp = 0.0
	if (t <= -6.8e+18)
		tmp = Float64(-t_1);
	elseif (t <= -2.85e-213)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(t_3 / x) + Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x)))))) / sqrt(2.0)));
	elseif (t <= 8e-206)
		tmp = Float64(Float64(t * sqrt(fma(x, 0.5, -0.5))) / Float64(l / sqrt(2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.8e+18], (-t$95$1), If[LessEqual[t, -2.85e-213], N[(t / N[(N[Sqrt[N[(N[(t$95$3 / x), $MachinePrecision] + N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-206], N[(N[(t * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.8e18

   1. Initial program 45.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 97.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.4%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 97.4%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 97.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -6.8e18 < t < -2.84999999999999997e-213

   1. Initial program 42.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around -inf 79.4%

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

   if -2.84999999999999997e-213 < t < 8.00000000000000023e-206

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 7.3%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.3%

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

      2. associate--l+ 35.4%

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

      3. sub-neg 35.4%

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

      4. metadata-eval 35.4%

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

      5. +-commutative 35.4%

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

      6. sub-neg 35.4%

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

      7. metadata-eval 35.4%

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

      8. +-commutative 35.4%

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

      9. associate-/l* 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in x around 0 49.7%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.6%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot x - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

      2. *-commutative 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot 0.5} - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      3. fma-neg 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      4. metadata-eval 49.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

   9. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

   if 8.00000000000000023e-206 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + \left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right)}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 3: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -8.8e+18)
     (- t_1)
     (if (<= t -2.75e-213)
       (/
        t
        (/
         (sqrt
          (+
           (+ (/ (pow l 2.0) x) (* 2.0 (* t (+ t (/ t x)))))
           (/ (fma 2.0 (pow t 2.0) (pow l 2.0)) x)))
         (sqrt 2.0)))
       (if (<= t 2.1e-206)
         (/ (* t (sqrt (fma x 0.5 -0.5))) (/ l (sqrt 2.0)))
         t_1)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -8.8e+18) {
		tmp = -t_1;
	} else if (t <= -2.75e-213) {
		tmp = t / (sqrt((((pow(l, 2.0) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, pow(t, 2.0), pow(l, 2.0)) / x))) / sqrt(2.0));
	} else if (t <= 2.1e-206) {
		tmp = (t * sqrt(fma(x, 0.5, -0.5))) / (l / sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -8.8e+18)
		tmp = Float64(-t_1);
	elseif (t <= -2.75e-213)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(Float64((l ^ 2.0) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, (t ^ 2.0), (l ^ 2.0)) / x))) / sqrt(2.0)));
	elseif (t <= 2.1e-206)
		tmp = Float64(Float64(t * sqrt(fma(x, 0.5, -0.5))) / Float64(l / sqrt(2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8.8e+18], (-t$95$1), If[LessEqual[t, -2.75e-213], N[(t / N[(N[Sqrt[N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e-206], N[(N[(t * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.8e18

   1. Initial program 45.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 97.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.4%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 97.4%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 97.4%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 97.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -8.8e18 < t < -2.75000000000000004e-213

   1. Initial program 42.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around inf 79.1%

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 77.9%

         \[\leadsto \frac{t}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}\right)\right)}}{\sqrt{2}}} \]

      2. expm1-udef 30.4%

         \[\leadsto \frac{t}{\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}\right)} - 1}}{\sqrt{2}}} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \frac{t}{\frac{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}\right)} - 1}}{\sqrt{2}}} \]
   7. Step-by-step derivation

      1. expm1-def 77.9%

         \[\leadsto \frac{t}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}\right)\right)}}{\sqrt{2}}} \]

      2. expm1-log1p 79.1%

         \[\leadsto \frac{t}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}}{\sqrt{2}}} \]

      3. fma-udef 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right)} + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      4. fma-udef 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \color{blue}{\left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      5. associate-+r+ 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      6. distribute-lft-out 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      7. unpow2 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      8. associate-*r/ 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\left(2 \cdot \left(\color{blue}{t \cdot \frac{t}{x}} + {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      9. unpow2 79.1%

         \[\leadsto \frac{t}{\frac{\sqrt{\left(2 \cdot \left(t \cdot \frac{t}{x} + \color{blue}{t \cdot t}\right) + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

      10. distribute-lft-out 79.1%

          \[\leadsto \frac{t}{\frac{\sqrt{\left(2 \cdot \color{blue}{\left(t \cdot \left(\frac{t}{x} + t\right)\right)} + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}} \]

   8. Simplified 79.1%

      \[\leadsto \frac{t}{\frac{\color{blue}{\sqrt{\left(2 \cdot \left(t \cdot \left(\frac{t}{x} + t\right)\right) + \frac{{\ell}^{2}}{x}\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}}{\sqrt{2}}} \]

   if -2.75000000000000004e-213 < t < 2.1000000000000001e-206

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 7.3%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.3%

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

      2. associate--l+ 35.4%

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

      3. sub-neg 35.4%

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

      4. metadata-eval 35.4%

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

      5. +-commutative 35.4%

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

      6. sub-neg 35.4%

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

      7. metadata-eval 35.4%

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

      8. +-commutative 35.4%

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

      9. associate-/l* 35.4%

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

   6. Simplified 35.4%

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

   7. Taylor expanded in x around 0 49.7%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.6%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot x - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

      2. *-commutative 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot 0.5} - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      3. fma-neg 49.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      4. metadata-eval 49.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

   9. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

   if 2.1000000000000001e-206 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-213}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\left(\frac{{\ell}^{2}}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 4: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-211}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-205}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -7e-211)
     (- t_1)
     (if (<= t 1.3e-205)
       (/ (* t (sqrt (fma x 0.5 -0.5))) (/ l (sqrt 2.0)))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -7e-211) {
		tmp = -t_1;
	} else if (t <= 1.3e-205) {
		tmp = (t * sqrt(fma(x, 0.5, -0.5))) / (l / sqrt(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -7e-211)
		tmp = Float64(-t_1);
	elseif (t <= 1.3e-205)
		tmp = Float64(Float64(t * sqrt(fma(x, 0.5, -0.5))) / Float64(l / sqrt(2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7e-211], (-t$95$1), If[LessEqual[t, 1.3e-205], N[(N[(t * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e-211

   1. Initial program 45.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 83.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 83.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 83.2%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 83.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -7e-211 < t < 1.2999999999999999e-205

   1. Initial program 3.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 6.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

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

      2. associate--l+ 35.9%

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

      3. sub-neg 35.9%

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

      4. metadata-eval 35.9%

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

      5. +-commutative 35.9%

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

      6. sub-neg 35.9%

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

      7. metadata-eval 35.9%

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

      8. +-commutative 35.9%

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

      9. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in x around 0 49.3%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 49.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5 \cdot x - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

      2. *-commutative 49.8%

         \[\leadsto \frac{\sqrt{\color{blue}{x \cdot 0.5} - 0.5} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      3. fma-neg 49.8%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

      4. metadata-eval 49.8%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot t}{\frac{\ell}{\sqrt{2}}} \]

   9. Applied egg-rr 49.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot t}{\frac{\ell}{\sqrt{2}}}} \]

   if 1.2999999999999999e-205 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-205}:\\ \;\;\;\;\frac{t \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -6.1 \cdot 10^{-211}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(t \cdot \frac{1}{\frac{\ell}{\sqrt{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -6.1e-211)
     (- t_1)
     (if (<= t 2.05e-206)
       (* (sqrt (- (* x 0.5) 0.5)) (* t (/ 1.0 (/ l (sqrt 2.0)))))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -6.1e-211) {
		tmp = -t_1;
	} else if (t <= 2.05e-206) {
		tmp = sqrt(((x * 0.5) - 0.5)) * (t * (1.0 / (l / sqrt(2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-6.1d-211)) then
        tmp = -t_1
    else if (t <= 2.05d-206) then
        tmp = sqrt(((x * 0.5d0) - 0.5d0)) * (t * (1.0d0 / (l / sqrt(2.0d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -6.1e-211) {
		tmp = -t_1;
	} else if (t <= 2.05e-206) {
		tmp = Math.sqrt(((x * 0.5) - 0.5)) * (t * (1.0 / (l / Math.sqrt(2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -6.1e-211:
		tmp = -t_1
	elif t <= 2.05e-206:
		tmp = math.sqrt(((x * 0.5) - 0.5)) * (t * (1.0 / (l / math.sqrt(2.0))))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -6.1e-211)
		tmp = Float64(-t_1);
	elseif (t <= 2.05e-206)
		tmp = Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(t * Float64(1.0 / Float64(l / sqrt(2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -6.1e-211)
		tmp = -t_1;
	elseif (t <= 2.05e-206)
		tmp = sqrt(((x * 0.5) - 0.5)) * (t * (1.0 / (l / sqrt(2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6.1e-211], (-t$95$1), If[LessEqual[t, 2.05e-206], N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(t * N[(1.0 / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.1e-211

   1. Initial program 45.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 83.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 83.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 83.2%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 83.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -6.1e-211 < t < 2.05000000000000008e-206

   1. Initial program 3.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 6.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

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

      2. associate--l+ 35.9%

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

      3. sub-neg 35.9%

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

      4. metadata-eval 35.9%

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

      5. +-commutative 35.9%

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

      6. sub-neg 35.9%

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

      7. metadata-eval 35.9%

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

      8. +-commutative 35.9%

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

      9. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in x around 0 49.3%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. div-inv 49.3%

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

   9. Applied egg-rr 49.3%

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

   if 2.05000000000000008e-206 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(t \cdot \frac{1}{\frac{\ell}{\sqrt{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(\sqrt{2} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -5.8e-211)
     (- t_1)
     (if (<= t 3.4e-206)
       (* (sqrt (- (* x 0.5) 0.5)) (* (sqrt 2.0) (/ t l)))
       t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -5.8e-211) {
		tmp = -t_1;
	} else if (t <= 3.4e-206) {
		tmp = sqrt(((x * 0.5) - 0.5)) * (sqrt(2.0) * (t / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-5.8d-211)) then
        tmp = -t_1
    else if (t <= 3.4d-206) then
        tmp = sqrt(((x * 0.5d0) - 0.5d0)) * (sqrt(2.0d0) * (t / l))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -5.8e-211) {
		tmp = -t_1;
	} else if (t <= 3.4e-206) {
		tmp = Math.sqrt(((x * 0.5) - 0.5)) * (Math.sqrt(2.0) * (t / l));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -5.8e-211:
		tmp = -t_1
	elif t <= 3.4e-206:
		tmp = math.sqrt(((x * 0.5) - 0.5)) * (math.sqrt(2.0) * (t / l))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -5.8e-211)
		tmp = Float64(-t_1);
	elseif (t <= 3.4e-206)
		tmp = Float64(sqrt(Float64(Float64(x * 0.5) - 0.5)) * Float64(sqrt(2.0) * Float64(t / l)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -5.8e-211)
		tmp = -t_1;
	elseif (t <= 3.4e-206)
		tmp = sqrt(((x * 0.5) - 0.5)) * (sqrt(2.0) * (t / l));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5.8e-211], (-t$95$1), If[LessEqual[t, 3.4e-206], N[(N[Sqrt[N[(N[(x * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.80000000000000029e-211

   1. Initial program 45.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 83.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 83.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 83.2%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 83.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -5.80000000000000029e-211 < t < 3.39999999999999985e-206

   1. Initial program 3.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 6.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

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

      2. associate--l+ 35.9%

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

      3. sub-neg 35.9%

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

      4. metadata-eval 35.9%

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

      5. +-commutative 35.9%

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

      6. sub-neg 35.9%

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

      7. metadata-eval 35.9%

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

      8. +-commutative 35.9%

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

      9. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in x around 0 49.3%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x - 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 49.3%

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

   9. Applied egg-rr 49.3%

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

   if 3.39999999999999985e-206 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0.5 - 0.5} \cdot \left(\sqrt{2} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 7: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-211}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{x \cdot 0.5} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -3.7e-211)
     (- t_1)
     (if (<= t 1.25e-205) (* (sqrt (* x 0.5)) (/ t (/ l (sqrt 2.0)))) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -3.7e-211) {
		tmp = -t_1;
	} else if (t <= 1.25e-205) {
		tmp = sqrt((x * 0.5)) * (t / (l / sqrt(2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-3.7d-211)) then
        tmp = -t_1
    else if (t <= 1.25d-205) then
        tmp = sqrt((x * 0.5d0)) * (t / (l / sqrt(2.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -3.7e-211) {
		tmp = -t_1;
	} else if (t <= 1.25e-205) {
		tmp = Math.sqrt((x * 0.5)) * (t / (l / Math.sqrt(2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -3.7e-211:
		tmp = -t_1
	elif t <= 1.25e-205:
		tmp = math.sqrt((x * 0.5)) * (t / (l / math.sqrt(2.0)))
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -3.7e-211)
		tmp = Float64(-t_1);
	elseif (t <= 1.25e-205)
		tmp = Float64(sqrt(Float64(x * 0.5)) * Float64(t / Float64(l / sqrt(2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -3.7e-211)
		tmp = -t_1;
	elseif (t <= 1.25e-205)
		tmp = sqrt((x * 0.5)) * (t / (l / sqrt(2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.7e-211], (-t$95$1), If[LessEqual[t, 1.25e-205], N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.6999999999999998e-211

   1. Initial program 45.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 83.2%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 83.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 83.2%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 83.2%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 83.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -3.6999999999999998e-211 < t < 1.25e-205

   1. Initial program 3.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   4. Taylor expanded in l around inf 6.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{\frac{1}{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

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

      2. associate--l+ 35.9%

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

      3. sub-neg 35.9%

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

      4. metadata-eval 35.9%

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

      5. +-commutative 35.9%

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

      6. sub-neg 35.9%

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

      7. metadata-eval 35.9%

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

      8. +-commutative 35.9%

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

      9. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in x around inf 45.2%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot x}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \sqrt{\color{blue}{x \cdot 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]

   9. Simplified 45.2%

      \[\leadsto \sqrt{\color{blue}{x \cdot 0.5}} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}} \]

   if 1.25e-205 < t

   1. Initial program 28.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 28.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 87.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-211}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{x \cdot 0.5} \cdot \frac{t}{\frac{\ell}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 8: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ -1.0 x) (+ x 1.0)))))
   (if (<= t -2e-311) (- t_1) t_1)))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -2e-311) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    if (t <= (-2d-311)) then
        tmp = -t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	double tmp;
	if (t <= -2e-311) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((-1.0 + x) / (x + 1.0)))
	tmp = 0
	if t <= -2e-311:
		tmp = -t_1
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2e-311)
		tmp = Float64(-t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((-1.0 + x) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -2e-311)
		tmp = -t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2e-311], (-t$95$1), t$95$1]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9999999999999e-311

   1. Initial program 40.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in t around -inf 75.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 75.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 75.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 75.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

      4. +-commutative 75.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{-1 + x}}{1 + x}} \]

   8. Simplified 75.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{1 + x}}} \]

   if -1.9999999999999e-311 < t

   1. Initial program 25.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 25.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 81.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 9: 76.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -8.5e-304) (/ t (- t)) (sqrt (/ (+ -1.0 x) (+ x 1.0)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.5e-304) {
		tmp = t / -t;
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.5d-304)) then
        tmp = t / -t
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.5e-304) {
		tmp = t / -t;
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -8.5e-304:
		tmp = t / -t
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -8.5e-304)
		tmp = Float64(t / Float64(-t));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -8.5e-304)
		tmp = t / -t;
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -8.5e-304], N[(t / (-t)), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.5e-304

   1. Initial program 40.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around inf 49.8%

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{\sqrt{2}}} \]

   5. Taylor expanded in t around -inf 75.1%

      \[\leadsto \frac{t}{\color{blue}{-1 \cdot t}} \]
   6. Step-by-step derivation

      1. neg-mul-1 75.1%

         \[\leadsto \frac{t}{\color{blue}{-t}} \]

   7. Simplified 75.1%

      \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -8.5e-304 < t

   1. Initial program 25.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 25.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   5. Applied egg-rr 68.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}{t}}} \]

   6. Taylor expanded in l around 0 80.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 10: 75.5% accurate, 37.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -8.5e-304) (/ t (- t)) (/ t t)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.5e-304) {
		tmp = t / -t;
	} else {
		tmp = t / t;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.5d-304)) then
        tmp = t / -t
    else
        tmp = t / t
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.5e-304) {
		tmp = t / -t;
	} else {
		tmp = t / t;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -8.5e-304:
		tmp = t / -t
	else:
		tmp = t / t
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -8.5e-304)
		tmp = Float64(t / Float64(-t));
	else
		tmp = Float64(t / t);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -8.5e-304)
		tmp = t / -t;
	else
		tmp = t / t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -8.5e-304], N[(t / (-t)), $MachinePrecision], N[(t / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.5e-304

   1. Initial program 40.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.6%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around inf 49.8%

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{\sqrt{2}}} \]

   5. Taylor expanded in t around -inf 75.1%

      \[\leadsto \frac{t}{\color{blue}{-1 \cdot t}} \]
   6. Step-by-step derivation

      1. neg-mul-1 75.1%

         \[\leadsto \frac{t}{\color{blue}{-t}} \]

   7. Simplified 75.1%

      \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -8.5e-304 < t

   1. Initial program 25.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

   4. Taylor expanded in x around inf 79.3%

      \[\leadsto \frac{t}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t}\\ \end{array} \]

Alternative 11: 38.8% accurate, 75.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{t}{t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (/ t t))

C

l = abs(l);
double code(double x, double l, double t) {
	return t / t;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = t / t
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return t / t;
}

Python

l = abs(l)
def code(x, l, t):
	return t / t

Julia

l = abs(l)
function code(x, l, t)
	return Float64(t / t)
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = t / t;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[(t / t), $MachinePrecision]

Derivation

1. Initial program 32.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 32.3%

   \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

4. Taylor expanded in x around inf 43.5%

   \[\leadsto \frac{t}{\color{blue}{t}} \]

5. Final simplification 43.5%

   \[\leadsto \frac{t}{t} \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))