Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 82.1%

Time: 31.8s

Alternatives: 14

Speedup: 225.0×

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: 33.3% 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: 82.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{t_1}, 2 \cdot \frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{x}^{4}}\right)\right)\right)}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t -1.05e+27)
     (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t 2.4e-302)
       (/
        1.0
        (/
         (sqrt
          (/
           (fma
            2.0
            (/ (* (+ x 1.0) (pow t 2.0)) (+ x -1.0))
            (* (/ 2.0 x) (pow l 2.0)))
           2.0))
         t))
       (if (<= t 7.8e-119)
         (/
          t
          (+
           t
           (*
            0.5
            (/ (* 2.0 (fma 2.0 (pow t 2.0) (pow l 2.0))) (* t (* x 2.0))))))
         (if (<= t 4.9e+143)
           (/
            t
            (/
             (sqrt
              (fma
               2.0
               (/ (pow t 2.0) t_1)
               (+
                (* 2.0 (/ (pow l 2.0) x))
                (+
                 (* 2.0 (/ (pow l 2.0) (pow x 2.0)))
                 (+
                  (* 2.0 (/ (pow l 2.0) (pow x 3.0)))
                  (* 2.0 (/ (pow l 2.0) (pow x 4.0))))))))
             (sqrt 2.0)))
           (sqrt t_1)))))))

C

double code(double x, double l, double t) {
	double t_1 = (x + -1.0) / (x + 1.0);
	double tmp;
	if (t <= -1.05e+27) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 2.4e-302) {
		tmp = 1.0 / (sqrt((fma(2.0, (((x + 1.0) * pow(t, 2.0)) / (x + -1.0)), ((2.0 / x) * pow(l, 2.0))) / 2.0)) / t);
	} else if (t <= 7.8e-119) {
		tmp = t / (t + (0.5 * ((2.0 * fma(2.0, pow(t, 2.0), pow(l, 2.0))) / (t * (x * 2.0)))));
	} else if (t <= 4.9e+143) {
		tmp = t / (sqrt(fma(2.0, (pow(t, 2.0) / t_1), ((2.0 * (pow(l, 2.0) / x)) + ((2.0 * (pow(l, 2.0) / pow(x, 2.0))) + ((2.0 * (pow(l, 2.0) / pow(x, 3.0))) + (2.0 * (pow(l, 2.0) / pow(x, 4.0)))))))) / sqrt(2.0));
	} else {
		tmp = sqrt(t_1);
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -1.05e+27)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 2.4e-302)
		tmp = Float64(1.0 / Float64(sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * (t ^ 2.0)) / Float64(x + -1.0)), Float64(Float64(2.0 / x) * (l ^ 2.0))) / 2.0)) / t));
	elseif (t <= 7.8e-119)
		tmp = Float64(t / Float64(t + Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t ^ 2.0), (l ^ 2.0))) / Float64(t * Float64(x * 2.0))))));
	elseif (t <= 4.9e+143)
		tmp = Float64(t / Float64(sqrt(fma(2.0, Float64((t ^ 2.0) / t_1), Float64(Float64(2.0 * Float64((l ^ 2.0) / x)) + Float64(Float64(2.0 * Float64((l ^ 2.0) / (x ^ 2.0))) + Float64(Float64(2.0 * Float64((l ^ 2.0) / (x ^ 3.0))) + Float64(2.0 * Float64((l ^ 2.0) / (x ^ 4.0)))))))) / sqrt(2.0)));
	else
		tmp = sqrt(t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.05e+27], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-302], N[(1.0 / N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-119], N[(t / N[(t + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.9e+143], N[(t / N[(N[Sqrt[N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.04999999999999997e27

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. +-commutative 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -1.04999999999999997e27 < t < 2.40000000000000022e-302

   1. Initial program 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in l around 0 49.1%

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

      1. fma-def 49.1%

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

      2. associate-/l* 49.1%

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

      3. +-commutative 49.1%

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

      4. sub-neg 49.1%

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

      5. metadata-eval 49.1%

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

      6. +-commutative 49.1%

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

      7. associate--l+ 60.2%

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

      8. sub-neg 60.2%

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

      9. metadata-eval 60.2%

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

      10. +-commutative 60.2%

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

      11. sub-neg 60.2%

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

      12. metadata-eval 60.2%

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

      13. +-commutative 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in x around inf 80.6%

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

      1. associate-*r/ 80.6%

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

   9. Simplified 80.6%

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

       1. clear-num 80.6%

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

       2. inv-pow 80.8%

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

       3. sqrt-undiv 81.0%

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

       4. associate-/r/ 64.4%

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

       5. associate-/l* 64.4%

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

   11. Applied egg-rr 64.4%

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

       1. unpow-1 64.4%

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

       2. associate-*l/ 80.9%

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

       3. +-commutative 80.9%

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

       4. +-commutative 80.9%

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

       5. associate-/r/ 80.9%

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

   13. Simplified 80.9%

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

   if 2.40000000000000022e-302 < t < 7.7999999999999998e-119

   1. Initial program 22.3%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around inf 73.3%

      \[\leadsto \frac{t}{\color{blue}{t + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 73.3%

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

      2. metadata-eval 73.3%

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

      3. distribute-rgt1-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. fma-def 73.3%

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

      6. *-commutative 73.3%

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

      7. unpow2 73.3%

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

      8. rem-square-sqrt 73.3%

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

   6. Simplified 73.3%

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

   if 7.7999999999999998e-119 < t < 4.89999999999999986e143

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in l around 0 63.5%

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

      1. fma-def 63.5%

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

      2. associate-/l* 76.1%

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

      3. +-commutative 76.1%

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

      4. sub-neg 76.1%

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

      5. metadata-eval 76.1%

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

      6. +-commutative 76.1%

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

      7. associate--l+ 78.1%

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

      8. sub-neg 78.1%

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

      9. metadata-eval 78.1%

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

      10. +-commutative 78.1%

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

      11. sub-neg 78.1%

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

      12. metadata-eval 78.1%

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

      13. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in x around inf 84.7%

      \[\leadsto \frac{t}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, \color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{x}^{4}}\right)\right)}\right)}}{\sqrt{2}}} \]

   if 4.89999999999999986e143 < t

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in l around 0 95.1%

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

      1. +-commutative 95.1%

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

      2. sub-neg 95.1%

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

      3. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in t around 0 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{x + -1}{x + 1}}, 2 \cdot \frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{x}^{4}}\right)\right)\right)}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{t_1}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t -1.55e+28)
     (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t 4.2e-303)
       (/
        1.0
        (/
         (sqrt
          (/
           (fma
            2.0
            (/ (* (+ x 1.0) (pow t 2.0)) (+ x -1.0))
            (* (/ 2.0 x) (pow l 2.0)))
           2.0))
         t))
       (if (<= t 7.8e-119)
         (/
          t
          (+
           t
           (*
            0.5
            (/ (* 2.0 (fma 2.0 (pow t 2.0) (pow l 2.0))) (* t (* x 2.0))))))
         (if (<= t 4.9e+143)
           (/
            t
            (/
             (sqrt (fma 2.0 (/ (pow t 2.0) t_1) (/ (* 2.0 (pow l 2.0)) x)))
             (sqrt 2.0)))
           (sqrt t_1)))))))

C

double code(double x, double l, double t) {
	double t_1 = (x + -1.0) / (x + 1.0);
	double tmp;
	if (t <= -1.55e+28) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 4.2e-303) {
		tmp = 1.0 / (sqrt((fma(2.0, (((x + 1.0) * pow(t, 2.0)) / (x + -1.0)), ((2.0 / x) * pow(l, 2.0))) / 2.0)) / t);
	} else if (t <= 7.8e-119) {
		tmp = t / (t + (0.5 * ((2.0 * fma(2.0, pow(t, 2.0), pow(l, 2.0))) / (t * (x * 2.0)))));
	} else if (t <= 4.9e+143) {
		tmp = t / (sqrt(fma(2.0, (pow(t, 2.0) / t_1), ((2.0 * pow(l, 2.0)) / x))) / sqrt(2.0));
	} else {
		tmp = sqrt(t_1);
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -1.55e+28)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 4.2e-303)
		tmp = Float64(1.0 / Float64(sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * (t ^ 2.0)) / Float64(x + -1.0)), Float64(Float64(2.0 / x) * (l ^ 2.0))) / 2.0)) / t));
	elseif (t <= 7.8e-119)
		tmp = Float64(t / Float64(t + Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t ^ 2.0), (l ^ 2.0))) / Float64(t * Float64(x * 2.0))))));
	elseif (t <= 4.9e+143)
		tmp = Float64(t / Float64(sqrt(fma(2.0, Float64((t ^ 2.0) / t_1), Float64(Float64(2.0 * (l ^ 2.0)) / x))) / sqrt(2.0)));
	else
		tmp = sqrt(t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.55e+28], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-303], N[(1.0 / N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-119], N[(t / N[(t + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.9e+143], N[(t / N[(N[Sqrt[N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[t$95$1], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.55e28

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. +-commutative 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -1.55e28 < t < 4.2e-303

   1. Initial program 42.8%

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

   3. Simplified 42.8%

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

   4. Taylor expanded in l around 0 49.1%

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

      1. fma-def 49.1%

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

      2. associate-/l* 49.1%

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

      3. +-commutative 49.1%

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

      4. sub-neg 49.1%

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

      5. metadata-eval 49.1%

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

      6. +-commutative 49.1%

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

      7. associate--l+ 60.2%

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

      8. sub-neg 60.2%

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

      9. metadata-eval 60.2%

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

      10. +-commutative 60.2%

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

      11. sub-neg 60.2%

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

      12. metadata-eval 60.2%

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

      13. +-commutative 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in x around inf 80.6%

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

      1. associate-*r/ 80.6%

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

   9. Simplified 80.6%

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

       1. clear-num 80.6%

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

       2. inv-pow 80.8%

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

       3. sqrt-undiv 81.0%

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

       4. associate-/r/ 64.4%

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

       5. associate-/l* 64.4%

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

   11. Applied egg-rr 64.4%

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

       1. unpow-1 64.4%

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

       2. associate-*l/ 80.9%

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

       3. +-commutative 80.9%

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

       4. +-commutative 80.9%

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

       5. associate-/r/ 80.9%

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

   13. Simplified 80.9%

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

   if 4.2e-303 < t < 7.7999999999999998e-119

   1. Initial program 22.3%

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

   3. Simplified 22.2%

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

   4. Taylor expanded in x around inf 73.3%

      \[\leadsto \frac{t}{\color{blue}{t + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 73.3%

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

      2. metadata-eval 73.3%

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

      3. distribute-rgt1-in 73.3%

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

      4. metadata-eval 73.3%

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

      5. fma-def 73.3%

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

      6. *-commutative 73.3%

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

      7. unpow2 73.3%

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

      8. rem-square-sqrt 73.3%

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

   6. Simplified 73.3%

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

   if 7.7999999999999998e-119 < t < 4.89999999999999986e143

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in l around 0 63.5%

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

      1. fma-def 63.5%

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

      2. associate-/l* 76.1%

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

      3. +-commutative 76.1%

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

      4. sub-neg 76.1%

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

      5. metadata-eval 76.1%

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

      6. +-commutative 76.1%

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

      7. associate--l+ 78.1%

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

      8. sub-neg 78.1%

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

      9. metadata-eval 78.1%

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

      10. +-commutative 78.1%

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

      11. sub-neg 78.1%

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

      12. metadata-eval 78.1%

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

      13. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in x around inf 83.7%

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

      1. associate-*r/ 83.7%

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

   9. Simplified 83.7%

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

   if 4.89999999999999986e143 < t

   1. Initial program 2.3%

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

   3. Simplified 2.3%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in l around 0 95.1%

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

      1. +-commutative 95.1%

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

      2. sub-neg 95.1%

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

      3. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in t around 0 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{x + -1}{x + 1}}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{if}\;t \leq -5.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (/
           (sqrt
            (/
             (fma
              2.0
              (/ (* (+ x 1.0) (pow t 2.0)) (+ x -1.0))
              (* (/ 2.0 x) (pow l 2.0)))
             2.0))
           t))))
   (if (<= t -5.7e+26)
     (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t 3.15e-304)
       t_1
       (if (<= t 6.8e-158)
         (/
          t
          (+
           t
           (*
            0.5
            (/ (* 2.0 (fma 2.0 (pow t 2.0) (pow l 2.0))) (* t (* x 2.0))))))
         (if (<= t 8.5e+23) t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = 1.0 / (sqrt((fma(2.0, (((x + 1.0) * pow(t, 2.0)) / (x + -1.0)), ((2.0 / x) * pow(l, 2.0))) / 2.0)) / t);
	double tmp;
	if (t <= -5.7e+26) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 3.15e-304) {
		tmp = t_1;
	} else if (t <= 6.8e-158) {
		tmp = t / (t + (0.5 * ((2.0 * fma(2.0, pow(t, 2.0), pow(l, 2.0))) / (t * (x * 2.0)))));
	} else if (t <= 8.5e+23) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(1.0 / Float64(sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * (t ^ 2.0)) / Float64(x + -1.0)), Float64(Float64(2.0 / x) * (l ^ 2.0))) / 2.0)) / t))
	tmp = 0.0
	if (t <= -5.7e+26)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 3.15e-304)
		tmp = t_1;
	elseif (t <= 6.8e-158)
		tmp = Float64(t / Float64(t + Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t ^ 2.0), (l ^ 2.0))) / Float64(t * Float64(x * 2.0))))));
	elseif (t <= 8.5e+23)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.7e+26], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.15e-304], t$95$1, If[LessEqual[t, 6.8e-158], N[(t / N[(t + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e+23], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.7000000000000003e26

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. +-commutative 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -5.7000000000000003e26 < t < 3.14999999999999992e-304 or 6.7999999999999999e-158 < t < 8.5000000000000001e23

   1. Initial program 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in l around 0 57.2%

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

      1. fma-def 57.2%

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

      2. associate-/l* 57.2%

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

      3. +-commutative 57.2%

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

      4. sub-neg 57.2%

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

      5. metadata-eval 57.2%

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

      6. +-commutative 57.2%

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

      7. associate--l+ 65.2%

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

      8. sub-neg 65.2%

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

      9. metadata-eval 65.2%

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

      10. +-commutative 65.2%

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

      11. sub-neg 65.2%

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

      12. metadata-eval 65.2%

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

      13. +-commutative 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in x around inf 81.0%

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

      1. associate-*r/ 81.0%

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

   9. Simplified 81.0%

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

       1. clear-num 81.0%

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

       2. inv-pow 81.3%

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

       3. sqrt-undiv 81.4%

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

       4. associate-/r/ 61.8%

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

       5. associate-/l* 61.8%

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

   11. Applied egg-rr 61.8%

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

       1. unpow-1 61.8%

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

       2. associate-*l/ 81.4%

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

       3. +-commutative 81.4%

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

       4. +-commutative 81.4%

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

       5. associate-/r/ 81.4%

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

   13. Simplified 81.4%

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

   if 3.14999999999999992e-304 < t < 6.7999999999999999e-158

   1. Initial program 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in x around inf 69.4%

      \[\leadsto \frac{t}{\color{blue}{t + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 69.4%

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

      2. metadata-eval 69.4%

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

      3. distribute-rgt1-in 69.4%

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

      4. metadata-eval 69.4%

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

      5. fma-def 69.4%

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

      6. *-commutative 69.4%

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

      7. unpow2 69.4%

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

      8. rem-square-sqrt 69.4%

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

   6. Simplified 69.4%

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

   if 8.5000000000000001e23 < t

   1. Initial program 31.5%

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

   3. Simplified 31.5%

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

   5. Applied egg-rr 32.2%

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

   6. Taylor expanded in l around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

   9. Taylor expanded in t around 0 90.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-301}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (fma
             2.0
             (/ (* (+ x 1.0) (pow t 2.0)) (+ x -1.0))
             (* (/ 2.0 x) (pow l 2.0)))
            2.0)))))
   (if (<= t -1.3e+31)
     (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t 1.22e-301)
       t_1
       (if (<= t 1.6e-157)
         (/
          t
          (+
           t
           (*
            0.5
            (/ (* 2.0 (fma 2.0 (pow t 2.0) (pow l 2.0))) (* t (* x 2.0))))))
         (if (<= t 1e+24) t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t / sqrt((fma(2.0, (((x + 1.0) * pow(t, 2.0)) / (x + -1.0)), ((2.0 / x) * pow(l, 2.0))) / 2.0));
	double tmp;
	if (t <= -1.3e+31) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 1.22e-301) {
		tmp = t_1;
	} else if (t <= 1.6e-157) {
		tmp = t / (t + (0.5 * ((2.0 * fma(2.0, pow(t, 2.0), pow(l, 2.0))) / (t * (x * 2.0)))));
	} else if (t <= 1e+24) {
		tmp = t_1;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * (t ^ 2.0)) / Float64(x + -1.0)), Float64(Float64(2.0 / x) * (l ^ 2.0))) / 2.0)))
	tmp = 0.0
	if (t <= -1.3e+31)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 1.22e-301)
		tmp = t_1;
	elseif (t <= 1.6e-157)
		tmp = Float64(t / Float64(t + Float64(0.5 * Float64(Float64(2.0 * fma(2.0, (t ^ 2.0), (l ^ 2.0))) / Float64(t * Float64(x * 2.0))))));
	elseif (t <= 1e+24)
		tmp = t_1;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.3e+31], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.22e-301], t$95$1, If[LessEqual[t, 1.6e-157], N[(t / N[(t + N[(0.5 * N[(N[(2.0 * N[(2.0 * N[Power[t, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+24], t$95$1, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.3e31

   1. Initial program 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in t around -inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. +-commutative 97.1%

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

      5. sub-neg 97.1%

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

      6. metadata-eval 97.1%

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

      7. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -1.3e31 < t < 1.2199999999999999e-301 or 1.6000000000000001e-157 < t < 9.9999999999999998e23

   1. Initial program 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in l around 0 57.2%

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

      1. fma-def 57.2%

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

      2. associate-/l* 57.2%

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

      3. +-commutative 57.2%

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

      4. sub-neg 57.2%

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

      5. metadata-eval 57.2%

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

      6. +-commutative 57.2%

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

      7. associate--l+ 65.2%

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

      8. sub-neg 65.2%

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

      9. metadata-eval 65.2%

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

      10. +-commutative 65.2%

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

      11. sub-neg 65.2%

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

      12. metadata-eval 65.2%

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

      13. +-commutative 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in x around inf 81.0%

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

      1. associate-*r/ 81.0%

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

   9. Simplified 81.0%

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

       1. expm1-log1p-u 79.9%

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

       2. expm1-udef 34.6%

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

       3. sqrt-undiv 34.6%

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

       4. associate-/r/ 34.6%

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

       5. associate-/l* 34.6%

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

   11. Applied egg-rr 34.6%

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

       1. expm1-def 60.6%

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

       2. expm1-log1p 61.8%

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

       3. associate-*l/ 81.4%

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

       4. +-commutative 81.4%

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

       5. +-commutative 81.4%

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

       6. associate-/r/ 81.4%

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

   13. Simplified 81.4%

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

   if 1.2199999999999999e-301 < t < 1.6000000000000001e-157

   1. Initial program 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in x around inf 69.4%

      \[\leadsto \frac{t}{\color{blue}{t + 0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 69.4%

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

      2. metadata-eval 69.4%

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

      3. distribute-rgt1-in 69.4%

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

      4. metadata-eval 69.4%

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

      5. fma-def 69.4%

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

      6. *-commutative 69.4%

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

      7. unpow2 69.4%

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

      8. rem-square-sqrt 69.4%

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

   6. Simplified 69.4%

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

   if 9.9999999999999998e23 < t

   1. Initial program 31.5%

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

   3. Simplified 31.5%

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

   5. Applied egg-rr 32.2%

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

   6. Taylor expanded in l around 0 90.1%

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

      1. +-commutative 90.1%

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

      2. sub-neg 90.1%

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

      3. metadata-eval 90.1%

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

   8. Simplified 90.1%

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

   9. Taylor expanded in t around 0 90.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-301}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{t}{t + 0.5 \cdot \frac{2 \cdot \mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{t \cdot \left(x \cdot 2\right)}}\\ \mathbf{elif}\;t \leq 10^{+24}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot {t}^{2}}{x + -1}, \frac{2}{x} \cdot {\ell}^{2}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-240}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{{x}^{2}} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.8e-190)
   (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
   (if (<= t 2.6e-240)
     (/
      t
      (*
       (/ l (sqrt 2.0))
       (sqrt (+ (/ 2.0 x) (+ (/ 2.0 (pow x 2.0)) (/ 2.0 (pow x 3.0)))))))
     (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-190) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 2.6e-240) {
		tmp = t / ((l / sqrt(2.0)) * sqrt(((2.0 / x) + ((2.0 / pow(x, 2.0)) + (2.0 / pow(x, 3.0))))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

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 <= (-1.8d-190)) then
        tmp = t / (t * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else if (t <= 2.6d-240) then
        tmp = t / ((l / sqrt(2.0d0)) * sqrt(((2.0d0 / x) + ((2.0d0 / (x ** 2.0d0)) + (2.0d0 / (x ** 3.0d0))))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-190) {
		tmp = t / (t * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 2.6e-240) {
		tmp = t / ((l / Math.sqrt(2.0)) * Math.sqrt(((2.0 / x) + ((2.0 / Math.pow(x, 2.0)) + (2.0 / Math.pow(x, 3.0))))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.8e-190:
		tmp = t / (t * -math.sqrt(((x + 1.0) / (x + -1.0))))
	elif t <= 2.6e-240:
		tmp = t / ((l / math.sqrt(2.0)) * math.sqrt(((2.0 / x) + ((2.0 / math.pow(x, 2.0)) + (2.0 / math.pow(x, 3.0))))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.8e-190)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 2.6e-240)
		tmp = Float64(t / Float64(Float64(l / sqrt(2.0)) * sqrt(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / (x ^ 2.0)) + Float64(2.0 / (x ^ 3.0)))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.8e-190)
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	elseif (t <= 2.6e-240)
		tmp = t / ((l / sqrt(2.0)) * sqrt(((2.0 / x) + ((2.0 / (x ^ 2.0)) + (2.0 / (x ^ 3.0))))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.8e-190], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-240], N[(t / N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.80000000000000003e-190

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around -inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

      4. +-commutative 85.6%

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

      5. sub-neg 85.6%

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

      6. metadata-eval 85.6%

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

      7. +-commutative 85.6%

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

   6. Simplified 85.6%

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

   if -1.80000000000000003e-190 < t < 2.59999999999999992e-240

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 4.8%

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

   5. Taylor expanded in x around inf 32.7%

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

      1. associate-+r+ 32.7%

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

      2. associate-*r/ 32.7%

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

      3. metadata-eval 32.7%

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

      4. associate-*r/ 32.7%

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

      5. metadata-eval 32.7%

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

      6. associate-*r/ 32.7%

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

      7. metadata-eval 32.7%

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

   7. Simplified 32.7%

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

   if 2.59999999999999992e-240 < t

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in l around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. sub-neg 82.3%

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

      3. metadata-eval 82.3%

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

   8. Simplified 82.3%

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

   9. Taylor expanded in t around 0 82.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-240}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{{x}^{2}} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-240}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x} + \frac{2}{{x}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-186)
   (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
   (if (<= t 1.05e-240)
     (/ t (* (/ l (sqrt 2.0)) (sqrt (+ (/ 2.0 x) (/ 2.0 (pow x 2.0))))))
     (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-186) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 1.05e-240) {
		tmp = t / ((l / sqrt(2.0)) * sqrt(((2.0 / x) + (2.0 / pow(x, 2.0)))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

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 <= (-2d-186)) then
        tmp = t / (t * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else if (t <= 1.05d-240) then
        tmp = t / ((l / sqrt(2.0d0)) * sqrt(((2.0d0 / x) + (2.0d0 / (x ** 2.0d0)))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-186) {
		tmp = t / (t * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 1.05e-240) {
		tmp = t / ((l / Math.sqrt(2.0)) * Math.sqrt(((2.0 / x) + (2.0 / Math.pow(x, 2.0)))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-186:
		tmp = t / (t * -math.sqrt(((x + 1.0) / (x + -1.0))))
	elif t <= 1.05e-240:
		tmp = t / ((l / math.sqrt(2.0)) * math.sqrt(((2.0 / x) + (2.0 / math.pow(x, 2.0)))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-186)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 1.05e-240)
		tmp = Float64(t / Float64(Float64(l / sqrt(2.0)) * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / (x ^ 2.0))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-186)
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	elseif (t <= 1.05e-240)
		tmp = t / ((l / sqrt(2.0)) * sqrt(((2.0 / x) + (2.0 / (x ^ 2.0)))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-186], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-240], N[(t / N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999998e-186

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around -inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

      4. +-commutative 85.6%

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

      5. sub-neg 85.6%

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

      6. metadata-eval 85.6%

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

      7. +-commutative 85.6%

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

   6. Simplified 85.6%

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

   if -1.9999999999999998e-186 < t < 1.04999999999999997e-240

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 4.8%

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

   5. Taylor expanded in x around inf 31.9%

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

      1. associate-*r/ 31.9%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{{x}^{2}}}} \]

      2. metadata-eval 31.9%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{{x}^{2}}}} \]

      3. associate-*r/ 31.9%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}}} \]

      4. metadata-eval 31.9%

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

   7. Simplified 31.9%

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

   if 1.04999999999999997e-240 < t

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 41.4%

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

   6. Taylor expanded in l around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. sub-neg 82.3%

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

      3. metadata-eval 82.3%

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

   8. Simplified 82.3%

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

   9. Taylor expanded in t around 0 82.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-240}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{2}{x} + \frac{2}{{x}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 76.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -8.8e-185)
   (/ t (* t (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
   (if (<= t 7e-300) (/ t (/ l (sqrt x))) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.8e-185) {
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 7e-300) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

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 <= (-8.8d-185)) then
        tmp = t / (t * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else if (t <= 7d-300) then
        tmp = t / (l / sqrt(x))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.8e-185) {
		tmp = t / (t * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= 7e-300) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -8.8e-185:
		tmp = t / (t * -math.sqrt(((x + 1.0) / (x + -1.0))))
	elif t <= 7e-300:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -8.8e-185)
		tmp = Float64(t / Float64(t * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= 7e-300)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -8.8e-185)
		tmp = t / (t * -sqrt(((x + 1.0) / (x + -1.0))));
	elseif (t <= 7e-300)
		tmp = t / (l / sqrt(x));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -8.8e-185], N[(t / N[(t * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-300], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.8000000000000002e-185

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around -inf 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

      4. +-commutative 85.6%

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

      5. sub-neg 85.6%

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

      6. metadata-eval 85.6%

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

      7. +-commutative 85.6%

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

   6. Simplified 85.6%

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

   if -8.8000000000000002e-185 < t < 7.0000000000000003e-300

   1. Initial program 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 1.5%

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

   5. Taylor expanded in x around inf 34.5%

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

      1. associate-*l/ 34.4%

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

      2. div-inv 34.4%

         \[\leadsto \frac{t}{\frac{\ell \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{x}}}}{\sqrt{2}}} \]

      3. sqrt-unprod 34.4%

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

      4. expm1-log1p-u 14.0%

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

      5. expm1-udef 7.3%

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

      6. associate-/l* 7.3%

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

      7. sqrt-unprod 7.3%

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

      8. div-inv 7.3%

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

      9. sqrt-undiv 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. expm1-def 14.1%

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

      2. expm1-log1p 34.5%

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

      3. associate-/r/ 34.5%

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

      4. metadata-eval 34.5%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{\color{blue}{1} \cdot x}}} \]

      5. *-lft-identity 34.5%

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

   9. Simplified 34.5%

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

   if 7.0000000000000003e-300 < t

   1. Initial program 37.7%

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

   3. Simplified 37.6%

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

   5. Applied egg-rr 38.2%

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

   6. Taylor expanded in l around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. sub-neg 79.4%

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

      3. metadata-eval 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in t around 0 79.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{t}{t \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -4.8e-189)
   (/ t (- (- t) (/ t x)))
   (if (<= t 7e-300) (/ t (/ l (sqrt x))) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.8e-189) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

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 <= (-4.8d-189)) then
        tmp = t / (-t - (t / x))
    else if (t <= 7d-300) then
        tmp = t / (l / sqrt(x))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.8e-189) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -4.8e-189:
		tmp = t / (-t - (t / x))
	elif t <= 7e-300:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -4.8e-189)
		tmp = Float64(t / Float64(Float64(-t) - Float64(t / x)));
	elseif (t <= 7e-300)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.8e-189)
		tmp = t / (-t - (t / x));
	elseif (t <= 7e-300)
		tmp = t / (l / sqrt(x));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -4.8e-189], N[(t / N[((-t) - N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-300], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.7999999999999997e-189

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Applied egg-rr 47.6%

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

   6. Taylor expanded in l around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 85.2%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t - \frac{t}{x}} \]

       7. mul-1-neg 85.2%

          \[\leadsto \frac{t}{\color{blue}{\left(-t\right)} - \frac{t}{x}} \]

   11. Simplified 85.2%

       \[\leadsto \frac{t}{\color{blue}{\left(-t\right) - \frac{t}{x}}} \]

   if -4.7999999999999997e-189 < t < 7.0000000000000003e-300

   1. Initial program 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 1.5%

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

   5. Taylor expanded in x around inf 34.5%

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

      1. associate-*l/ 34.4%

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

      2. div-inv 34.4%

         \[\leadsto \frac{t}{\frac{\ell \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{x}}}}{\sqrt{2}}} \]

      3. sqrt-unprod 34.4%

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

      4. expm1-log1p-u 14.0%

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

      5. expm1-udef 7.3%

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

      6. associate-/l* 7.3%

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

      7. sqrt-unprod 7.3%

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

      8. div-inv 7.3%

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

      9. sqrt-undiv 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. expm1-def 14.1%

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

      2. expm1-log1p 34.5%

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

      3. associate-/r/ 34.5%

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

      4. metadata-eval 34.5%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{\color{blue}{1} \cdot x}}} \]

      5. *-lft-identity 34.5%

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

   9. Simplified 34.5%

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

   if 7.0000000000000003e-300 < t

   1. Initial program 37.7%

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

   3. Simplified 37.6%

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

   5. Applied egg-rr 38.2%

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

   6. Taylor expanded in l around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. sub-neg 79.4%

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

      3. metadata-eval 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in t around 0 79.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9: 76.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -4.8e-187)
   (/ t (- (- t) (/ t x)))
   (if (<= t 7e-300) (* (sqrt x) (/ t l)) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.8e-187) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

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 <= (-4.8d-187)) then
        tmp = t / (-t - (t / x))
    else if (t <= 7d-300) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.8e-187) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -4.8e-187:
		tmp = t / (-t - (t / x))
	elif t <= 7e-300:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -4.8e-187)
		tmp = Float64(t / Float64(Float64(-t) - Float64(t / x)));
	elseif (t <= 7e-300)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.8e-187)
		tmp = t / (-t - (t / x));
	elseif (t <= 7e-300)
		tmp = sqrt(x) * (t / l);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -4.8e-187], N[(t / N[((-t) - N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-300], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.80000000000000027e-187

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Applied egg-rr 47.6%

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

   6. Taylor expanded in l around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 85.2%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t - \frac{t}{x}} \]

       7. mul-1-neg 85.2%

          \[\leadsto \frac{t}{\color{blue}{\left(-t\right)} - \frac{t}{x}} \]

   11. Simplified 85.2%

       \[\leadsto \frac{t}{\color{blue}{\left(-t\right) - \frac{t}{x}}} \]

   if -4.80000000000000027e-187 < t < 7.0000000000000003e-300

   1. Initial program 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 1.5%

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

   5. Taylor expanded in x around inf 34.5%

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

   6. Taylor expanded in t around 0 34.6%

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

   if 7.0000000000000003e-300 < t

   1. Initial program 37.7%

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

   3. Simplified 37.6%

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

   5. Applied egg-rr 38.2%

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

   6. Taylor expanded in l around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. sub-neg 79.4%

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

      3. metadata-eval 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in x around inf 79.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-189}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -3.15e-189)
   (/ t (- (- t) (/ t x)))
   (if (<= t 7e-300) (/ t (/ l (sqrt x))) (+ 1.0 (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.15e-189) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = t / (l / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

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 <= (-3.15d-189)) then
        tmp = t / (-t - (t / x))
    else if (t <= 7d-300) then
        tmp = t / (l / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.15e-189) {
		tmp = t / (-t - (t / x));
	} else if (t <= 7e-300) {
		tmp = t / (l / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -3.15e-189:
		tmp = t / (-t - (t / x))
	elif t <= 7e-300:
		tmp = t / (l / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -3.15e-189)
		tmp = Float64(t / Float64(Float64(-t) - Float64(t / x)));
	elseif (t <= 7e-300)
		tmp = Float64(t / Float64(l / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.15e-189)
		tmp = t / (-t - (t / x));
	elseif (t <= 7e-300)
		tmp = t / (l / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -3.15e-189], N[(t / N[((-t) - N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-300], N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.15e-189

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Applied egg-rr 47.6%

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

   6. Taylor expanded in l around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 85.2%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t - \frac{t}{x}} \]

       7. mul-1-neg 85.2%

          \[\leadsto \frac{t}{\color{blue}{\left(-t\right)} - \frac{t}{x}} \]

   11. Simplified 85.2%

       \[\leadsto \frac{t}{\color{blue}{\left(-t\right) - \frac{t}{x}}} \]

   if -3.15e-189 < t < 7.0000000000000003e-300

   1. Initial program 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 1.5%

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

   5. Taylor expanded in x around inf 34.5%

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

      1. associate-*l/ 34.4%

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

      2. div-inv 34.4%

         \[\leadsto \frac{t}{\frac{\ell \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{x}}}}{\sqrt{2}}} \]

      3. sqrt-unprod 34.4%

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

      4. expm1-log1p-u 14.0%

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

      5. expm1-udef 7.3%

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

      6. associate-/l* 7.3%

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

      7. sqrt-unprod 7.3%

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

      8. div-inv 7.3%

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

      9. sqrt-undiv 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. expm1-def 14.1%

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

      2. expm1-log1p 34.5%

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

      3. associate-/r/ 34.5%

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

      4. metadata-eval 34.5%

         \[\leadsto \frac{t}{\frac{\ell}{\sqrt{\color{blue}{1} \cdot x}}} \]

      5. *-lft-identity 34.5%

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

   9. Simplified 34.5%

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

   if 7.0000000000000003e-300 < t

   1. Initial program 37.7%

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

   3. Simplified 37.6%

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

   5. Applied egg-rr 38.2%

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

   6. Taylor expanded in l around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. sub-neg 79.4%

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

      3. metadata-eval 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in x around inf 79.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-189}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-300}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11: 76.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (/ t (- (- t) (/ t x))) (+ 1.0 (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = t / (-t - (t / x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

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 <= (-5d-310)) then
        tmp = t / (-t - (t / x))
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = t / (-t - (t / x));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = t / (-t - (t / x))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(t / Float64(Float64(-t) - Float64(t / x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = t / (-t - (t / x));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(t / N[((-t) - N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 42.0%

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

   3. Simplified 42.0%

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

   5. Applied egg-rr 42.7%

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

   6. Taylor expanded in l around 0 1.8%

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

      1. +-commutative 1.8%

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

      2. sub-neg 1.8%

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

      3. metadata-eval 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 77.3%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t - \frac{t}{x}} \]

       7. mul-1-neg 77.3%

          \[\leadsto \frac{t}{\color{blue}{\left(-t\right)} - \frac{t}{x}} \]

   11. Simplified 77.3%

       \[\leadsto \frac{t}{\color{blue}{\left(-t\right) - \frac{t}{x}}} \]

   if -4.999999999999985e-310 < t

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   5. Applied egg-rr 37.6%

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

   6. Taylor expanded in l around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. sub-neg 78.2%

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

      3. metadata-eval 78.2%

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

   8. Simplified 78.2%

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

   9. Taylor expanded in x around inf 77.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{\left(-t\right) - \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12: 75.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -1.2e-279) (/ t (- t)) (+ 1.0 (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.2e-279) {
		tmp = t / -t;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

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 <= (-1.2d-279)) then
        tmp = t / -t
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.2e-279) {
		tmp = t / -t;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.2e-279:
		tmp = t / -t
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.2e-279)
		tmp = Float64(t / Float64(-t));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.2e-279)
		tmp = t / -t;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.2e-279], N[(t / (-t)), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999995e-279

   1. Initial program 42.3%

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

   3. Simplified 42.3%

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

   5. Applied egg-rr 43.0%

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

   6. Taylor expanded in l around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 76.7%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       4. mul-1-neg 76.7%

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

   11. Simplified 76.7%

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

   if -1.19999999999999995e-279 < t

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   5. Applied egg-rr 37.3%

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

   6. Taylor expanded in l around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. sub-neg 77.6%

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

      3. metadata-eval 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in x around inf 77.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13: 75.0% accurate, 37.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -1.2e-279) (/ t (- t)) 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.2e-279) {
		tmp = t / -t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 <= (-1.2d-279)) then
        tmp = t / -t
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.2e-279) {
		tmp = t / -t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -1.2e-279:
		tmp = t / -t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -1.2e-279)
		tmp = Float64(t / Float64(-t));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.2e-279)
		tmp = t / -t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -1.2e-279], N[(t / (-t)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999995e-279

   1. Initial program 42.3%

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

   3. Simplified 42.3%

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

   5. Applied egg-rr 43.0%

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

   6. Taylor expanded in l around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in x around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 76.7%

          \[\leadsto \frac{t}{\color{blue}{-1} \cdot t} \]

       4. mul-1-neg 76.7%

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

   11. Simplified 76.7%

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

   if -1.19999999999999995e-279 < t

   1. Initial program 36.8%

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

   3. Simplified 36.8%

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

   5. Applied egg-rr 37.3%

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

   6. Taylor expanded in l around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. sub-neg 77.6%

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

      3. metadata-eval 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in x around inf 77.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 38.9% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 1.0)

C

double code(double x, double l, double t) {
	return 1.0;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

public static double code(double x, double l, double t) {
	return 1.0;
}

Python

def code(x, l, t):
	return 1.0

Julia

function code(x, l, t)
	return 1.0
end

MATLAB

function tmp = code(x, l, t)
	tmp = 1.0;
end

Wolfram

code[x_, l_, t_] := 1.0

Derivation

1. Initial program 39.6%

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

3. Simplified 39.6%

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

5. Applied egg-rr 40.2%

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

6. Taylor expanded in l around 0 39.1%

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

   1. +-commutative 39.1%

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

   2. sub-neg 39.1%

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

   3. metadata-eval 39.1%

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

8. Simplified 39.1%

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

9. Taylor expanded in x around inf 38.8%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 38.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023333 
(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)))))