Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 83.4%

Time: 17.1s

Alternatives: 20

Speedup: 13.2×

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: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{t_1}{x}\right)}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-217}:\\ \;\;\;\;\frac{t_2}{t_3}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left|t_2\right|}{\mathsf{fma}\left(0.5, \frac{t_1 + t_1}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_2\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (fma 2.0 (* t t) (* l l)))
        (t_2 (* t (sqrt 2.0)))
        (t_3
         (sqrt
          (+ (/ (* l l) x) (+ (* 2.0 (+ (* t t) (/ (* t t) x))) (/ t_1 x))))))
   (if (<= t -3.8e+37)
     (cbrt
      (pow (* (sqrt 2.0) (- (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))) 3.0))
     (if (<= t -4.2e-217)
       (/ t_2 t_3)
       (if (<= t 3.8e-162)
         (/ (fabs t_2) (fma 0.5 (/ (+ t_1 t_1) (* (sqrt 2.0) (* t x))) t_2))
         (if (<= t 2.7e+91) (/ (sqrt (* t (* t 2.0))) t_3) (/ t_2 t_2)))))))

C

double code(double x, double l, double t) {
	double t_1 = fma(2.0, (t * t), (l * l));
	double t_2 = t * sqrt(2.0);
	double t_3 = sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (t_1 / x))));
	double tmp;
	if (t <= -3.8e+37) {
		tmp = cbrt(pow((sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))))), 3.0));
	} else if (t <= -4.2e-217) {
		tmp = t_2 / t_3;
	} else if (t <= 3.8e-162) {
		tmp = fabs(t_2) / fma(0.5, ((t_1 + t_1) / (sqrt(2.0) * (t * x))), t_2);
	} else if (t <= 2.7e+91) {
		tmp = sqrt((t * (t * 2.0))) / t_3;
	} else {
		tmp = t_2 / t_2;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = fma(2.0, Float64(t * t), Float64(l * l))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(t_1 / x))))
	tmp = 0.0
	if (t <= -3.8e+37)
		tmp = cbrt((Float64(sqrt(2.0) * Float64(-sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))))) ^ 3.0));
	elseif (t <= -4.2e-217)
		tmp = Float64(t_2 / t_3);
	elseif (t <= 3.8e-162)
		tmp = Float64(abs(t_2) / fma(0.5, Float64(Float64(t_1 + t_1) / Float64(sqrt(2.0) * Float64(t * x))), t_2));
	elseif (t <= 2.7e+91)
		tmp = Float64(sqrt(Float64(t * Float64(t * 2.0))) / t_3);
	else
		tmp = Float64(t_2 / t_2);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.8e+37], N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, -4.2e-217], N[(t$95$2 / t$95$3), $MachinePrecision], If[LessEqual[t, 3.8e-162], N[(N[Abs[t$95$2], $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 + t$95$1), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e+91], N[(N[Sqrt[N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$2 / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.7999999999999999e37

   1. Initial program 33.5%

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

   2. Taylor expanded in t around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. associate-*l* 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. +-commutative 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   4. Simplified 94.2%

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

      1. add-cbrt-cube 95.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right)}} \]

      2. cube-unmult 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   if -3.7999999999999999e37 < t < -4.2e-217

   1. Initial program 33.2%

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

   2. Taylor expanded in x around inf 72.5%

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

      1. associate--l+ 72.5%

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

      2. unpow2 72.5%

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

      3. distribute-lft-out 72.5%

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

      4. unpow2 72.5%

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

      5. unpow2 72.5%

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

      6. associate-*r/ 72.5%

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

      7. mul-1-neg 72.5%

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

      8. +-commutative 72.5%

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

      9. unpow2 72.5%

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

      10. unpow2 72.5%

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

      11. fma-udef 72.5%

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

   4. Simplified 72.5%

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

   if -4.2e-217 < t < 3.80000000000000005e-162

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 52.0%

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

      1. fma-def 52.0%

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

      2. +-commutative 52.0%

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

      3. unpow2 52.0%

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

      4. unpow2 52.0%

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

      5. fma-udef 52.0%

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

      6. mul-1-neg 52.0%

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

      7. +-commutative 52.0%

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

      8. unpow2 52.0%

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

      9. unpow2 52.0%

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

      10. fma-udef 52.0%

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

      11. *-commutative 52.0%

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

   4. Simplified 52.0%

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

      1. add-sqr-sqrt 34.8%

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

      2. sqrt-prod 32.2%

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

      3. sqrt-prod 32.2%

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

      4. add-sqr-sqrt 32.2%

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

      5. rem-sqrt-square 32.2%

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

      6. sqrt-prod 32.2%

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

      7. sqrt-prod 34.8%

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

      8. add-sqr-sqrt 70.8%

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

      9. *-commutative 70.8%

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

   6. Applied egg-rr 70.8%

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

   if 3.80000000000000005e-162 < t < 2.7e91

   1. Initial program 65.1%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 98.9%

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

      2. unpow2 98.9%

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

      3. distribute-lft-out 98.9%

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

      4. unpow2 98.9%

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

      5. unpow2 98.9%

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

      6. associate-*r/ 98.9%

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

      7. mul-1-neg 98.9%

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

      8. +-commutative 98.9%

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

      9. unpow2 98.9%

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

      10. unpow2 98.9%

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

      11. fma-udef 98.9%

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

   4. Simplified 98.9%

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

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-prod 99.5%

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

      3. sqrt-prod 100.0%

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

      4. pow1/2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1/2 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   if 2.7e91 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2} \cdot t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-217}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left|t \cdot \sqrt{2}\right|}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 2: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{t_1}{t_2}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(0.5, 2 \cdot \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_1\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2
         (sqrt
          (+
           (/ (* l l) x)
           (+
            (* 2.0 (+ (* t t) (/ (* t t) x)))
            (/ (fma 2.0 (* t t) (* l l)) x))))))
   (if (<= t -2e+37)
     (cbrt
      (pow (* (sqrt 2.0) (- (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))) 3.0))
     (if (<= t -1.8e-293)
       (/ t_1 t_2)
       (if (<= t 3.8e-162)
         (/ t_1 (fma 0.5 (* 2.0 (/ (* l l) (* (sqrt 2.0) (* t x)))) t_1))
         (if (<= t 5.2e+90) (/ (sqrt (* t (* t 2.0))) t_2) (/ t_1 t_1)))))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x))));
	double tmp;
	if (t <= -2e+37) {
		tmp = cbrt(pow((sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))))), 3.0));
	} else if (t <= -1.8e-293) {
		tmp = t_1 / t_2;
	} else if (t <= 3.8e-162) {
		tmp = t_1 / fma(0.5, (2.0 * ((l * l) / (sqrt(2.0) * (t * x)))), t_1);
	} else if (t <= 5.2e+90) {
		tmp = sqrt((t * (t * 2.0))) / t_2;
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))
	tmp = 0.0
	if (t <= -2e+37)
		tmp = cbrt((Float64(sqrt(2.0) * Float64(-sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))))) ^ 3.0));
	elseif (t <= -1.8e-293)
		tmp = Float64(t_1 / t_2);
	elseif (t <= 3.8e-162)
		tmp = Float64(t_1 / fma(0.5, Float64(2.0 * Float64(Float64(l * l) / Float64(sqrt(2.0) * Float64(t * x)))), t_1));
	elseif (t <= 5.2e+90)
		tmp = Float64(sqrt(Float64(t * Float64(t * 2.0))) / t_2);
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2e+37], N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, -1.8e-293], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[t, 3.8e-162], N[(t$95$1 / N[(0.5 * N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.2e+90], N[(N[Sqrt[N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.99999999999999991e37

   1. Initial program 33.5%

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

   2. Taylor expanded in t around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. associate-*l* 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. +-commutative 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   4. Simplified 94.2%

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

      1. add-cbrt-cube 95.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right)}} \]

      2. cube-unmult 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   if -1.99999999999999991e37 < t < -1.79999999999999993e-293

   1. Initial program 29.2%

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

   2. Taylor expanded in x around inf 66.5%

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

      1. associate--l+ 66.5%

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

      2. unpow2 66.5%

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

      3. distribute-lft-out 66.5%

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

      4. unpow2 66.5%

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

      5. unpow2 66.5%

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

      6. associate-*r/ 66.5%

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

      7. mul-1-neg 66.5%

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

      8. +-commutative 66.5%

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

      9. unpow2 66.5%

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

      10. unpow2 66.5%

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

      11. fma-udef 66.5%

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

   4. Simplified 66.5%

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

   if -1.79999999999999993e-293 < t < 3.80000000000000005e-162

   1. Initial program 7.3%

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

   2. Taylor expanded in x around inf 60.5%

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

      1. fma-def 60.5%

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

      2. +-commutative 60.5%

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

      3. unpow2 60.5%

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

      4. unpow2 60.5%

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

      5. fma-udef 60.5%

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

      6. mul-1-neg 60.5%

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

      7. +-commutative 60.5%

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

      8. unpow2 60.5%

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

      9. unpow2 60.5%

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

      10. fma-udef 60.5%

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

      11. *-commutative 60.5%

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

   4. Simplified 60.5%

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

   5. Taylor expanded in t around 0 60.5%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \color{blue}{2 \cdot \frac{{\ell}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)}}, t \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 60.5%

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

   7. Simplified 60.5%

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

   if 3.80000000000000005e-162 < t < 5.1999999999999997e90

   1. Initial program 65.1%

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

   2. Taylor expanded in x around inf 98.9%

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

      1. associate--l+ 98.9%

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

      2. unpow2 98.9%

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

      3. distribute-lft-out 98.9%

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

      4. unpow2 98.9%

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

      5. unpow2 98.9%

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

      6. associate-*r/ 98.9%

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

      7. mul-1-neg 98.9%

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

      8. +-commutative 98.9%

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

      9. unpow2 98.9%

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

      10. unpow2 98.9%

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

      11. fma-udef 98.9%

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

   4. Simplified 98.9%

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

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-prod 99.5%

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

      3. sqrt-prod 100.0%

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

      4. pow1/2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1/2 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   if 5.1999999999999997e90 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2} \cdot t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, 2 \cdot \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 3: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -1e+39)
     (cbrt
      (pow (* (sqrt 2.0) (- (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))) 3.0))
     (if (<= t 6.4e+90)
       (/
        t_1
        (sqrt
         (+
          (/ (* l l) x)
          (+
           (* 2.0 (+ (* t t) (/ (* t t) x)))
           (/ (fma 2.0 (* t t) (* l l)) x)))))
       (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -1e+39) {
		tmp = cbrt(pow((sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))))), 3.0));
	} else if (t <= 6.4e+90) {
		tmp = t_1 / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -1e+39)
		tmp = cbrt((Float64(sqrt(2.0) * Float64(-sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))))) ^ 3.0));
	elseif (t <= 6.4e+90)
		tmp = Float64(t_1 / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+39], N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, 6.4e+90], N[(t$95$1 / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.9999999999999994e38

   1. Initial program 33.5%

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

   2. Taylor expanded in t around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. associate-*l* 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. +-commutative 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   4. Simplified 94.2%

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

      1. add-cbrt-cube 95.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right)}} \]

      2. cube-unmult 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   if -9.9999999999999994e38 < t < 6.39999999999999997e90

   1. Initial program 41.5%

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

   2. Taylor expanded in x around inf 76.5%

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

      1. associate--l+ 76.5%

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

      2. unpow2 76.5%

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

      3. distribute-lft-out 76.5%

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

      4. unpow2 76.5%

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

      5. unpow2 76.5%

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

      6. associate-*r/ 76.5%

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

      7. mul-1-neg 76.5%

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

      8. +-commutative 76.5%

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

      9. unpow2 76.5%

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

      10. unpow2 76.5%

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

      11. fma-udef 76.5%

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

   4. Simplified 76.5%

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

   if 6.39999999999999997e90 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 4: 81.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5\right)\right)}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -2.5e+38)
     (cbrt (* (* 2.0 (sqrt 2.0)) (* (sqrt 0.5) (- 0.5))))
     (if (<= t 5.3e+90)
       (/
        t_1
        (sqrt
         (+
          (/ (* l l) x)
          (+
           (* 2.0 (+ (* t t) (/ (* t t) x)))
           (/ (fma 2.0 (* t t) (* l l)) x)))))
       (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -2.5e+38) {
		tmp = cbrt(((2.0 * sqrt(2.0)) * (sqrt(0.5) * -0.5)));
	} else if (t <= 5.3e+90) {
		tmp = t_1 / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -2.5e+38)
		tmp = cbrt(Float64(Float64(2.0 * sqrt(2.0)) * Float64(sqrt(0.5) * Float64(-0.5))));
	elseif (t <= 5.3e+90)
		tmp = Float64(t_1 / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.5e+38], N[Power[N[(N[(2.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, 5.3e+90], N[(t$95$1 / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.49999999999999985e38

   1. Initial program 33.5%

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

   2. Taylor expanded in t around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. associate-*l* 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. +-commutative 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   4. Simplified 94.2%

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

      1. add-cbrt-cube 95.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right)}} \]

      2. cube-unmult 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   9. Taylor expanded in x around inf 94.9%

      \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \left({\left(\sqrt{2}\right)}^{3} \cdot {\left(\sqrt{0.5}\right)}^{3}\right)}} \]
   10. Step-by-step derivation

       1. mul-1-neg 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{-{\left(\sqrt{2}\right)}^{3} \cdot {\left(\sqrt{0.5}\right)}^{3}}} \]

       2. distribute-rgt-neg-in 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2}\right)}^{3} \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)}} \]

       3. unpow3 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)} \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)} \]

       4. rem-square-sqrt 94.9%

          \[\leadsto \sqrt[3]{\left(\color{blue}{2} \cdot \sqrt{2}\right) \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)} \]

       5. unpow3 94.9%

          \[\leadsto \sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(-\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \sqrt{0.5}}\right)} \]

       6. rem-square-sqrt 94.9%

          \[\leadsto \sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(-\color{blue}{0.5} \cdot \sqrt{0.5}\right)} \]

   11. Simplified 94.9%

       \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot \sqrt{2}\right) \cdot \left(-0.5 \cdot \sqrt{0.5}\right)}} \]

   if -2.49999999999999985e38 < t < 5.29999999999999979e90

   1. Initial program 41.5%

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

   2. Taylor expanded in x around inf 76.5%

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

      1. associate--l+ 76.5%

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

      2. unpow2 76.5%

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

      3. distribute-lft-out 76.5%

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

      4. unpow2 76.5%

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

      5. unpow2 76.5%

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

      6. associate-*r/ 76.5%

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

      7. mul-1-neg 76.5%

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

      8. +-commutative 76.5%

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

      9. unpow2 76.5%

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

      10. unpow2 76.5%

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

      11. fma-udef 76.5%

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

   4. Simplified 76.5%

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

   if 5.29999999999999979e90 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5\right)\right)}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 5: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))) (t_2 (/ (* l l) x)))
   (if (<= t -1e+38)
     (cbrt (* (* 2.0 (sqrt 2.0)) (* (sqrt 0.5) (- 0.5))))
     (if (<= t 8.5e+16)
       (/ t_1 (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x)))))))
       (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = (l * l) / x;
	double tmp;
	if (t <= -1e+38) {
		tmp = cbrt(((2.0 * sqrt(2.0)) * (sqrt(0.5) * -0.5)));
	} else if (t <= 8.5e+16) {
		tmp = t_1 / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double t_2 = (l * l) / x;
	double tmp;
	if (t <= -1e+38) {
		tmp = Math.cbrt(((2.0 * Math.sqrt(2.0)) * (Math.sqrt(0.5) * -0.5)));
	} else if (t <= 8.5e+16) {
		tmp = t_1 / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -1e+38)
		tmp = cbrt(Float64(Float64(2.0 * sqrt(2.0)) * Float64(sqrt(0.5) * Float64(-0.5))));
	elseif (t <= 8.5e+16)
		tmp = Float64(t_1 / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1e+38], N[Power[N[(N[(2.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[t, 8.5e+16], N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.99999999999999977e37

   1. Initial program 33.5%

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

   2. Taylor expanded in t around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. associate-*l* 94.2%

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

      3. distribute-rgt-neg-in 94.2%

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

      4. +-commutative 94.2%

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

      5. sub-neg 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   4. Simplified 94.2%

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

      1. add-cbrt-cube 95.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   6. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*l* 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)\right)}} \]

      2. cube-unmult 95.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   8. Simplified 95.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}^{3}}} \]

   9. Taylor expanded in x around inf 94.9%

      \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \left({\left(\sqrt{2}\right)}^{3} \cdot {\left(\sqrt{0.5}\right)}^{3}\right)}} \]
   10. Step-by-step derivation

       1. mul-1-neg 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{-{\left(\sqrt{2}\right)}^{3} \cdot {\left(\sqrt{0.5}\right)}^{3}}} \]

       2. distribute-rgt-neg-in 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2}\right)}^{3} \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)}} \]

       3. unpow3 94.9%

          \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)} \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)} \]

       4. rem-square-sqrt 94.9%

          \[\leadsto \sqrt[3]{\left(\color{blue}{2} \cdot \sqrt{2}\right) \cdot \left(-{\left(\sqrt{0.5}\right)}^{3}\right)} \]

       5. unpow3 94.9%

          \[\leadsto \sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(-\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{0.5}\right) \cdot \sqrt{0.5}}\right)} \]

       6. rem-square-sqrt 94.9%

          \[\leadsto \sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(-\color{blue}{0.5} \cdot \sqrt{0.5}\right)} \]

   11. Simplified 94.9%

       \[\leadsto \sqrt[3]{\color{blue}{\left(2 \cdot \sqrt{2}\right) \cdot \left(-0.5 \cdot \sqrt{0.5}\right)}} \]

   if -9.99999999999999977e37 < t < 8.5e16

   1. Initial program 33.5%

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

   2. Taylor expanded in x around inf 72.7%

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

      1. associate--l+ 72.7%

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

      2. unpow2 72.7%

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

      3. distribute-lft-out 72.7%

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

      4. unpow2 72.7%

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

      5. unpow2 72.7%

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

      6. associate-*r/ 72.7%

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

      7. mul-1-neg 72.7%

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

      8. +-commutative 72.7%

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

      9. unpow2 72.7%

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

      10. unpow2 72.7%

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

      11. fma-udef 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. mul-1-neg 72.6%

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

      3. unpow2 72.6%

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

      4. distribute-rgt-neg-in 72.6%

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

   7. Simplified 72.6%

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

   if 8.5e16 < t

   1. Initial program 45.5%

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

   2. Taylor expanded in x around inf 96.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+38}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{2}\right) \cdot \left(\sqrt{0.5} \cdot \left(-0.5\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))) (t_2 (/ (* l l) x)))
   (if (<= t -6.5e+50)
     (* (sqrt 2.0) (- (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))
     (if (<= t 2.1e+15)
       (/ t_1 (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x)))))))
       (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = (l * l) / x;
	double tmp;
	if (t <= -6.5e+50) {
		tmp = sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	} else if (t <= 2.1e+15) {
		tmp = t_1 / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))));
	} else {
		tmp = t_1 / t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    t_2 = (l * l) / x
    if (t <= (-6.5d+50)) then
        tmp = sqrt(2.0d0) * -sqrt((0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0))))
    else if (t <= 2.1d+15) then
        tmp = t_1 / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x))))))
    else
        tmp = t_1 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double t_2 = (l * l) / x;
	double tmp;
	if (t <= -6.5e+50) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	} else if (t <= 2.1e+15) {
		tmp = t_1 / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	t_2 = (l * l) / x
	tmp = 0
	if t <= -6.5e+50:
		tmp = math.sqrt(2.0) * -math.sqrt((0.5 * ((x + -1.0) / (x + 1.0))))
	elif t <= 2.1e+15:
		tmp = t_1 / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))))
	else:
		tmp = t_1 / t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -6.5e+50)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0))))));
	elseif (t <= 2.1e+15)
		tmp = Float64(t_1 / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	t_2 = (l * l) / x;
	tmp = 0.0;
	if (t <= -6.5e+50)
		tmp = sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	elseif (t <= 2.1e+15)
		tmp = t_1 / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x))))));
	else
		tmp = t_1 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -6.5e+50], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 2.1e+15], N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5000000000000003e50

   1. Initial program 30.3%

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

   2. Taylor expanded in t around -inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. associate-*l* 94.0%

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

      3. distribute-rgt-neg-in 94.0%

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

      4. +-commutative 94.0%

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

      5. sub-neg 94.0%

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

      6. metadata-eval 94.0%

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

      7. +-commutative 94.0%

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

   4. Simplified 94.0%

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

      1. pow1 94.0%

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

      2. sqrt-unprod 94.0%

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

      3. +-commutative 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. unpow1 94.0%

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

   8. Simplified 94.0%

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

   if -6.5000000000000003e50 < t < 2.1e15

   1. Initial program 35.1%

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

   2. Taylor expanded in x around inf 73.4%

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

      1. associate--l+ 73.4%

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

      2. unpow2 73.4%

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

      3. distribute-lft-out 73.4%

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

      4. unpow2 73.4%

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

      5. unpow2 73.4%

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

      6. associate-*r/ 73.4%

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

      7. mul-1-neg 73.4%

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

      8. +-commutative 73.4%

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

      9. unpow2 73.4%

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

      10. unpow2 73.4%

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

      11. fma-udef 73.4%

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

   4. Simplified 73.4%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. mul-1-neg 73.2%

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

      3. unpow2 73.2%

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

      4. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

   if 2.1e15 < t

   1. Initial program 45.5%

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

   2. Taylor expanded in x around inf 96.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 7: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -2.35e-147)
     (* (sqrt 2.0) (* (sqrt 0.5) (- -1.0 (/ -1.0 x))))
     (if (<= t 4.4e-119) (/ t_1 (sqrt (* 2.0 (/ l (/ x l))))) (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -2.35e-147) {
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 4.4e-119) {
		tmp = t_1 / sqrt((2.0 * (l / (x / l))));
	} else {
		tmp = t_1 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if (t <= (-2.35d-147)) then
        tmp = sqrt(2.0d0) * (sqrt(0.5d0) * ((-1.0d0) - ((-1.0d0) / x)))
    else if (t <= 4.4d-119) then
        tmp = t_1 / sqrt((2.0d0 * (l / (x / l))))
    else
        tmp = t_1 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if (t <= -2.35e-147) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 4.4e-119) {
		tmp = t_1 / Math.sqrt((2.0 * (l / (x / l))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if t <= -2.35e-147:
		tmp = math.sqrt(2.0) * (math.sqrt(0.5) * (-1.0 - (-1.0 / x)))
	elif t <= 4.4e-119:
		tmp = t_1 / math.sqrt((2.0 * (l / (x / l))))
	else:
		tmp = t_1 / t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -2.35e-147)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(0.5) * Float64(-1.0 - Float64(-1.0 / x))));
	elseif (t <= 4.4e-119)
		tmp = Float64(t_1 / sqrt(Float64(2.0 * Float64(l / Float64(x / l)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if (t <= -2.35e-147)
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	elseif (t <= 4.4e-119)
		tmp = t_1 / sqrt((2.0 * (l / (x / l))));
	else
		tmp = t_1 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.35e-147], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.4e-119], N[(t$95$1 / N[Sqrt[N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.34999999999999994e-147

   1. Initial program 38.6%

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

   2. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. associate-*l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. +-commutative 85.7%

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

      5. sub-neg 85.7%

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

      6. metadata-eval 85.7%

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

      7. +-commutative 85.7%

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

   4. Simplified 85.7%

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

   5. Taylor expanded in x around inf 85.1%

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

      1. pow1 85.1%

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

      2. distribute-rgt-neg-in 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. unpow1 85.1%

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

      2. neg-sub0 85.1%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(0 - \left(1 - \frac{1}{x}\right)\right)}\right) \]

      3. associate--r- 85.1%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(\left(0 - 1\right) + \frac{1}{x}\right)}\right) \]

      4. metadata-eval 85.1%

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

   9. Simplified 85.1%

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

   if -2.34999999999999994e-147 < t < 4.4000000000000001e-119

   1. Initial program 9.6%

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

   2. Taylor expanded in x around inf 54.0%

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

      1. associate--l+ 54.0%

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

      2. unpow2 54.0%

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

      3. distribute-lft-out 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. mul-1-neg 54.0%

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

      8. +-commutative 54.0%

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

      9. unpow2 54.0%

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

      10. unpow2 54.0%

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

      11. fma-udef 54.0%

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

   4. Simplified 54.0%

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

      1. add-cube-cbrt 53.9%

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

      2. associate-/l* 54.0%

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

      3. associate-/l* 54.0%

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

      4. associate-/l* 53.9%

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

   6. Applied egg-rr 53.9%

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

   7. Taylor expanded in t around 0 50.7%

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

      1. cancel-sign-sub-inv 50.7%

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

      2. unpow2 50.7%

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

      3. associate-*l/ 50.7%

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

      4. metadata-eval 50.7%

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

      5. unpow2 50.7%

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

      6. associate-*l/ 54.1%

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

      7. distribute-rgt1-in 54.1%

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

      8. metadata-eval 54.1%

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

      9. associate-*l/ 50.7%

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

      10. associate-/l* 54.0%

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

   9. Simplified 54.0%

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

   if 4.4000000000000001e-119 < t

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 93.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 8: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -1.2e-145)
     (* (sqrt 2.0) (- (sqrt (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))
     (if (<= t 3.5e-119) (/ t_1 (sqrt (* 2.0 (/ l (/ x l))))) (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -1.2e-145) {
		tmp = sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	} else if (t <= 3.5e-119) {
		tmp = t_1 / sqrt((2.0 * (l / (x / l))));
	} else {
		tmp = t_1 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if (t <= (-1.2d-145)) then
        tmp = sqrt(2.0d0) * -sqrt((0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0))))
    else if (t <= 3.5d-119) then
        tmp = t_1 / sqrt((2.0d0 * (l / (x / l))))
    else
        tmp = t_1 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if (t <= -1.2e-145) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	} else if (t <= 3.5e-119) {
		tmp = t_1 / Math.sqrt((2.0 * (l / (x / l))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if t <= -1.2e-145:
		tmp = math.sqrt(2.0) * -math.sqrt((0.5 * ((x + -1.0) / (x + 1.0))))
	elif t <= 3.5e-119:
		tmp = t_1 / math.sqrt((2.0 * (l / (x / l))))
	else:
		tmp = t_1 / t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -1.2e-145)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0))))));
	elseif (t <= 3.5e-119)
		tmp = Float64(t_1 / sqrt(Float64(2.0 * Float64(l / Float64(x / l)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if (t <= -1.2e-145)
		tmp = sqrt(2.0) * -sqrt((0.5 * ((x + -1.0) / (x + 1.0))));
	elseif (t <= 3.5e-119)
		tmp = t_1 / sqrt((2.0 * (l / (x / l))));
	else
		tmp = t_1 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e-145], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.5e-119], N[(t$95$1 / N[Sqrt[N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.20000000000000008e-145

   1. Initial program 38.6%

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

   2. Taylor expanded in t around -inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. associate-*l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

      4. +-commutative 85.7%

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

      5. sub-neg 85.7%

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

      6. metadata-eval 85.7%

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

      7. +-commutative 85.7%

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

   4. Simplified 85.7%

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

      1. pow1 85.7%

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

      2. sqrt-unprod 85.7%

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

      3. +-commutative 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. unpow1 85.7%

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

   8. Simplified 85.7%

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

   if -1.20000000000000008e-145 < t < 3.5e-119

   1. Initial program 9.6%

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

   2. Taylor expanded in x around inf 54.0%

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

      1. associate--l+ 54.0%

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

      2. unpow2 54.0%

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

      3. distribute-lft-out 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. mul-1-neg 54.0%

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

      8. +-commutative 54.0%

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

      9. unpow2 54.0%

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

      10. unpow2 54.0%

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

      11. fma-udef 54.0%

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

   4. Simplified 54.0%

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

      1. add-cube-cbrt 53.9%

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

      2. associate-/l* 54.0%

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

      3. associate-/l* 54.0%

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

      4. associate-/l* 53.9%

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

   6. Applied egg-rr 53.9%

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

   7. Taylor expanded in t around 0 50.7%

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

      1. cancel-sign-sub-inv 50.7%

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

      2. unpow2 50.7%

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

      3. associate-*l/ 50.7%

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

      4. metadata-eval 50.7%

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

      5. unpow2 50.7%

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

      6. associate-*l/ 54.1%

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

      7. distribute-rgt1-in 54.1%

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

      8. metadata-eval 54.1%

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

      9. associate-*l/ 50.7%

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

      10. associate-/l* 54.0%

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

   9. Simplified 54.0%

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

   if 3.5e-119 < t

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 93.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 9: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{t_1}{\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -5.2e-209)
     (* (sqrt 2.0) (* (sqrt 0.5) (- -1.0 (/ -1.0 x))))
     (if (<= t 3.8e-171) (/ t_1 (* (- l) (sqrt (/ 2.0 x)))) (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -5.2e-209) {
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 3.8e-171) {
		tmp = t_1 / (-l * sqrt((2.0 / x)));
	} else {
		tmp = t_1 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if (t <= (-5.2d-209)) then
        tmp = sqrt(2.0d0) * (sqrt(0.5d0) * ((-1.0d0) - ((-1.0d0) / x)))
    else if (t <= 3.8d-171) then
        tmp = t_1 / (-l * sqrt((2.0d0 / x)))
    else
        tmp = t_1 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if (t <= -5.2e-209) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 3.8e-171) {
		tmp = t_1 / (-l * Math.sqrt((2.0 / x)));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if t <= -5.2e-209:
		tmp = math.sqrt(2.0) * (math.sqrt(0.5) * (-1.0 - (-1.0 / x)))
	elif t <= 3.8e-171:
		tmp = t_1 / (-l * math.sqrt((2.0 / x)))
	else:
		tmp = t_1 / t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -5.2e-209)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(0.5) * Float64(-1.0 - Float64(-1.0 / x))));
	elseif (t <= 3.8e-171)
		tmp = Float64(t_1 / Float64(Float64(-l) * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if (t <= -5.2e-209)
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	elseif (t <= 3.8e-171)
		tmp = t_1 / (-l * sqrt((2.0 / x)));
	else
		tmp = t_1 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e-209], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-171], N[(t$95$1 / N[((-l) * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.19999999999999969e-209

   1. Initial program 34.8%

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

   2. Taylor expanded in t around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. associate-*l* 80.5%

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

      3. distribute-rgt-neg-in 80.5%

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

      4. +-commutative 80.5%

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

      5. sub-neg 80.5%

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

      6. metadata-eval 80.5%

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

      7. +-commutative 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in x around inf 80.0%

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

      1. pow1 80.0%

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

      2. distribute-rgt-neg-in 80.0%

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

   7. Applied egg-rr 80.0%

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

      1. unpow1 80.0%

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

      2. neg-sub0 80.0%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(0 - \left(1 - \frac{1}{x}\right)\right)}\right) \]

      3. associate--r- 80.0%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(\left(0 - 1\right) + \frac{1}{x}\right)}\right) \]

      4. metadata-eval 80.0%

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

   9. Simplified 80.0%

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

   if -5.19999999999999969e-209 < t < 3.80000000000000021e-171

   1. Initial program 7.9%

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

   2. Taylor expanded in l around -inf 8.5%

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

      1. associate-*r* 8.5%

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

      2. neg-mul-1 8.5%

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

      3. sub-neg 8.5%

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

      4. sub-neg 8.5%

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

      5. metadata-eval 8.5%

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

      6. +-commutative 8.5%

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

      7. sub-neg 8.5%

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

      8. metadata-eval 8.5%

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

      9. +-commutative 8.5%

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

      10. metadata-eval 8.5%

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

   4. Simplified 8.5%

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

   5. Taylor expanded in x around inf 52.9%

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

   if 3.80000000000000021e-171 < t

   1. Initial program 47.7%

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

   2. Taylor expanded in x around inf 90.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-\ell\right) \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 10: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+90}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -9.5e-238)
     (* (sqrt 2.0) (- (sqrt 0.5)))
     (if (<= t 3.5e-119)
       (* (/ t l) (sqrt x))
       (if (<= t 5.5e+90)
         (+ 1.0 (* -0.5 (/ (fma 2.0 (* t t) (* l l)) (* x (* t t)))))
         (/ t_1 t_1))))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -9.5e-238) {
		tmp = sqrt(2.0) * -sqrt(0.5);
	} else if (t <= 3.5e-119) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= 5.5e+90) {
		tmp = 1.0 + (-0.5 * (fma(2.0, (t * t), (l * l)) / (x * (t * t))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -9.5e-238)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(0.5)));
	elseif (t <= 3.5e-119)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= 5.5e+90)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(fma(2.0, Float64(t * t), Float64(l * l)) / Float64(x * Float64(t * t)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e-238], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.5e-119], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+90], N[(1.0 + N[(-0.5 * N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.50000000000000059e-238

   1. Initial program 32.1%

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

   2. Taylor expanded in t around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-*l* 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. sub-neg 76.8%

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

      6. metadata-eval 76.8%

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

      7. +-commutative 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in x around inf 75.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 75.8%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{0.5}} \]

      2. distribute-rgt-neg-in 75.8%

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

   7. Simplified 75.8%

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

   if -9.50000000000000059e-238 < t < 3.5e-119

   1. Initial program 13.6%

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

   2. Taylor expanded in l around inf 8.3%

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

   3. Taylor expanded in x around inf 52.0%

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

   4. Taylor expanded in t around 0 47.1%

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

   if 3.5e-119 < t < 5.49999999999999999e90

   1. Initial program 69.1%

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

   2. Taylor expanded in x around inf 91.5%

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

      1. fma-def 91.5%

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

      2. +-commutative 91.5%

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

      3. unpow2 91.5%

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

      4. unpow2 91.5%

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

      5. fma-udef 91.5%

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

      6. mul-1-neg 91.5%

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

      7. +-commutative 91.5%

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

      8. unpow2 91.5%

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

      9. unpow2 91.5%

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

      10. fma-udef 91.5%

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

      11. *-commutative 91.5%

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

   4. Simplified 91.5%

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

   5. Taylor expanded in x around -inf 90.9%

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

      1. associate-*r/ 90.9%

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

      2. unpow2 90.9%

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

      3. rem-square-sqrt 90.9%

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

      4. times-frac 90.9%

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

      5. metadata-eval 90.9%

         \[\leadsto 1 + \color{blue}{-0.5} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{t}^{2} \cdot x} \]

      6. +-commutative 90.9%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{{t}^{2} \cdot x} \]

      7. unpow2 90.9%

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

      8. unpow2 90.9%

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

      9. fma-udef 90.9%

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

      10. unpow2 90.9%

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

   7. Simplified 90.9%

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

   if 5.49999999999999999e90 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2} \cdot t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+90}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 11: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 10^{+91}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -2.9e-238)
     (* (sqrt 2.0) (* (sqrt 0.5) (- -1.0 (/ -1.0 x))))
     (if (<= t 3.5e-119)
       (* (/ t l) (sqrt x))
       (if (<= t 1e+91)
         (+ 1.0 (* -0.5 (/ (fma 2.0 (* t t) (* l l)) (* x (* t t)))))
         (/ t_1 t_1))))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -2.9e-238) {
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 3.5e-119) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= 1e+91) {
		tmp = 1.0 + (-0.5 * (fma(2.0, (t * t), (l * l)) / (x * (t * t))));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -2.9e-238)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(0.5) * Float64(-1.0 - Float64(-1.0 / x))));
	elseif (t <= 3.5e-119)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= 1e+91)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(fma(2.0, Float64(t * t), Float64(l * l)) / Float64(x * Float64(t * t)))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.9e-238], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-119], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+91], N[(1.0 + N[(-0.5 * N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(x * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.8999999999999998e-238

   1. Initial program 32.1%

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

   2. Taylor expanded in t around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-*l* 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. sub-neg 76.8%

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

      6. metadata-eval 76.8%

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

      7. +-commutative 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in x around inf 76.4%

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

      1. pow1 76.4%

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

      2. distribute-rgt-neg-in 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. unpow1 76.4%

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

      2. neg-sub0 76.4%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(0 - \left(1 - \frac{1}{x}\right)\right)}\right) \]

      3. associate--r- 76.4%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(\left(0 - 1\right) + \frac{1}{x}\right)}\right) \]

      4. metadata-eval 76.4%

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

   9. Simplified 76.4%

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

   if -2.8999999999999998e-238 < t < 3.5e-119

   1. Initial program 13.6%

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

   2. Taylor expanded in l around inf 8.3%

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

   3. Taylor expanded in x around inf 52.0%

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

   4. Taylor expanded in t around 0 47.1%

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

   if 3.5e-119 < t < 1.00000000000000008e91

   1. Initial program 69.1%

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

   2. Taylor expanded in x around inf 91.5%

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

      1. fma-def 91.5%

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

      2. +-commutative 91.5%

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

      3. unpow2 91.5%

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

      4. unpow2 91.5%

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

      5. fma-udef 91.5%

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

      6. mul-1-neg 91.5%

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

      7. +-commutative 91.5%

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

      8. unpow2 91.5%

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

      9. unpow2 91.5%

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

      10. fma-udef 91.5%

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

      11. *-commutative 91.5%

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

   4. Simplified 91.5%

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

   5. Taylor expanded in x around -inf 90.9%

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

      1. associate-*r/ 90.9%

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

      2. unpow2 90.9%

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

      3. rem-square-sqrt 90.9%

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

      4. times-frac 90.9%

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

      5. metadata-eval 90.9%

         \[\leadsto 1 + \color{blue}{-0.5} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{t}^{2} \cdot x} \]

      6. +-commutative 90.9%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{{t}^{2} \cdot x} \]

      7. unpow2 90.9%

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

      8. unpow2 90.9%

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

      9. fma-udef 90.9%

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

      10. unpow2 90.9%

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

   7. Simplified 90.9%

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

   if 1.00000000000000008e91 < t

   1. Initial program 27.7%

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

   2. Taylor expanded in x around inf 96.3%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2} \cdot t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 10^{+91}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 12: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -2.95 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t_1}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= t -2.95e-238)
     (* (sqrt 2.0) (* (sqrt 0.5) (- -1.0 (/ -1.0 x))))
     (if (<= t 3.5e-119) (/ t_1 (* l (sqrt (/ 2.0 x)))) (/ t_1 t_1)))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (t <= -2.95e-238) {
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 3.5e-119) {
		tmp = t_1 / (l * sqrt((2.0 / x)));
	} else {
		tmp = t_1 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t * sqrt(2.0d0)
    if (t <= (-2.95d-238)) then
        tmp = sqrt(2.0d0) * (sqrt(0.5d0) * ((-1.0d0) - ((-1.0d0) / x)))
    else if (t <= 3.5d-119) then
        tmp = t_1 / (l * sqrt((2.0d0 / x)))
    else
        tmp = t_1 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if (t <= -2.95e-238) {
		tmp = Math.sqrt(2.0) * (Math.sqrt(0.5) * (-1.0 - (-1.0 / x)));
	} else if (t <= 3.5e-119) {
		tmp = t_1 / (l * Math.sqrt((2.0 / x)));
	} else {
		tmp = t_1 / t_1;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if t <= -2.95e-238:
		tmp = math.sqrt(2.0) * (math.sqrt(0.5) * (-1.0 - (-1.0 / x)))
	elif t <= 3.5e-119:
		tmp = t_1 / (l * math.sqrt((2.0 / x)))
	else:
		tmp = t_1 / t_1
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -2.95e-238)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(0.5) * Float64(-1.0 - Float64(-1.0 / x))));
	elseif (t <= 3.5e-119)
		tmp = Float64(t_1 / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(t_1 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if (t <= -2.95e-238)
		tmp = sqrt(2.0) * (sqrt(0.5) * (-1.0 - (-1.0 / x)));
	elseif (t <= 3.5e-119)
		tmp = t_1 / (l * sqrt((2.0 / x)));
	else
		tmp = t_1 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.95e-238], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-119], N[(t$95$1 / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.9499999999999999e-238

   1. Initial program 32.1%

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

   2. Taylor expanded in t around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-*l* 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. sub-neg 76.8%

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

      6. metadata-eval 76.8%

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

      7. +-commutative 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in x around inf 76.4%

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

      1. pow1 76.4%

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

      2. distribute-rgt-neg-in 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. unpow1 76.4%

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

      2. neg-sub0 76.4%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(0 - \left(1 - \frac{1}{x}\right)\right)}\right) \]

      3. associate--r- 76.4%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(\left(0 - 1\right) + \frac{1}{x}\right)}\right) \]

      4. metadata-eval 76.4%

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

   9. Simplified 76.4%

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

   if -2.9499999999999999e-238 < t < 3.5e-119

   1. Initial program 13.6%

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

   2. Taylor expanded in l around inf 8.3%

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

   3. Taylor expanded in x around inf 52.0%

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

   if 3.5e-119 < t

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 93.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-1 - \frac{-1}{x}\right)\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2}}\\ \end{array} \]

Alternative 13: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+174}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -9e-238)
   (* (sqrt 2.0) (- (sqrt 0.5)))
   (if (<= t 3.9e-119)
     (* (/ t l) (sqrt x))
     (if (<= t 1.05e+174)
       (- 1.0 (/ (+ 1.0 (/ (* l l) (* 2.0 (* t t)))) x))
       (* (sqrt 2.0) (sqrt 0.5))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -9e-238) {
		tmp = sqrt(2.0) * -sqrt(0.5);
	} else if (t <= 3.9e-119) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= 1.05e+174) {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	} else {
		tmp = sqrt(2.0) * sqrt(0.5);
	}
	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 <= (-9d-238)) then
        tmp = sqrt(2.0d0) * -sqrt(0.5d0)
    else if (t <= 3.9d-119) then
        tmp = (t / l) * sqrt(x)
    else if (t <= 1.05d+174) then
        tmp = 1.0d0 - ((1.0d0 + ((l * l) / (2.0d0 * (t * t)))) / x)
    else
        tmp = sqrt(2.0d0) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -9e-238) {
		tmp = Math.sqrt(2.0) * -Math.sqrt(0.5);
	} else if (t <= 3.9e-119) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= 1.05e+174) {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	} else {
		tmp = Math.sqrt(2.0) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -9e-238:
		tmp = math.sqrt(2.0) * -math.sqrt(0.5)
	elif t <= 3.9e-119:
		tmp = (t / l) * math.sqrt(x)
	elif t <= 1.05e+174:
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x)
	else:
		tmp = math.sqrt(2.0) * math.sqrt(0.5)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -9e-238)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(0.5)));
	elseif (t <= 3.9e-119)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= 1.05e+174)
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(l * l) / Float64(2.0 * Float64(t * t)))) / x));
	else
		tmp = Float64(sqrt(2.0) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -9e-238)
		tmp = sqrt(2.0) * -sqrt(0.5);
	elseif (t <= 3.9e-119)
		tmp = (t / l) * sqrt(x);
	elseif (t <= 1.05e+174)
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	else
		tmp = sqrt(2.0) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -9e-238], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.9e-119], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e+174], N[(1.0 - N[(N[(1.0 + N[(N[(l * l), $MachinePrecision] / N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.99999999999999992e-238

   1. Initial program 32.1%

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

   2. Taylor expanded in t around -inf 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-*l* 76.8%

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

      3. distribute-rgt-neg-in 76.8%

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

      4. +-commutative 76.8%

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

      5. sub-neg 76.8%

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

      6. metadata-eval 76.8%

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

      7. +-commutative 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in x around inf 75.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 75.8%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{0.5}} \]

      2. distribute-rgt-neg-in 75.8%

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

   7. Simplified 75.8%

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

   if -8.99999999999999992e-238 < t < 3.8999999999999999e-119

   1. Initial program 13.6%

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

   2. Taylor expanded in l around inf 8.3%

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

   3. Taylor expanded in x around inf 52.0%

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

   4. Taylor expanded in t around 0 47.1%

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

   if 3.8999999999999999e-119 < t < 1.05000000000000008e174

   1. Initial program 64.7%

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

   2. Taylor expanded in x around inf 81.5%

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

      1. fma-def 81.5%

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

      2. +-commutative 81.5%

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

      3. unpow2 81.5%

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

      4. unpow2 81.5%

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

      5. fma-udef 81.5%

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

      6. mul-1-neg 81.5%

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

      7. +-commutative 81.5%

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

      8. unpow2 81.5%

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

      9. unpow2 81.5%

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

      10. fma-udef 81.5%

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

      11. *-commutative 81.5%

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

   4. Simplified 81.5%

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

   5. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. unsub-neg 91.1%

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

   7. Simplified 91.1%

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

   if 1.05000000000000008e174 < t

   1. Initial program 2.6%

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

   2. Taylor expanded in x around inf 98.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+174}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 14: 44.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 3.5e-119) (* (/ t l) (sqrt x)) (* (sqrt 2.0) (sqrt 0.5))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 3.5e-119) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = sqrt(2.0) * sqrt(0.5);
	}
	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 <= 3.5d-119) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = sqrt(2.0d0) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 3.5e-119) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = Math.sqrt(2.0) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 3.5e-119:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = math.sqrt(2.0) * math.sqrt(0.5)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 3.5e-119)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = Float64(sqrt(2.0) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 3.5e-119)
		tmp = (t / l) * sqrt(x);
	else
		tmp = sqrt(2.0) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 3.5e-119], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e-119

   1. Initial program 28.3%

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

   2. Taylor expanded in l around inf 3.4%

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

   3. Taylor expanded in x around inf 19.5%

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

   4. Taylor expanded in t around 0 18.0%

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

   if 3.5e-119 < t

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 91.8%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 15: 40.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 3.5e-119)
   (* (/ t l) (sqrt x))
   (- 1.0 (/ (+ 1.0 (/ (* l l) (* 2.0 (* t t)))) x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 3.5e-119) {
		tmp = (t / l) * sqrt(x);
	} else {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / 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 <= 3.5d-119) then
        tmp = (t / l) * sqrt(x)
    else
        tmp = 1.0d0 - ((1.0d0 + ((l * l) / (2.0d0 * (t * t)))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 3.5e-119) {
		tmp = (t / l) * Math.sqrt(x);
	} else {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 3.5e-119:
		tmp = (t / l) * math.sqrt(x)
	else:
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 3.5e-119)
		tmp = Float64(Float64(t / l) * sqrt(x));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(l * l) / Float64(2.0 * Float64(t * t)))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 3.5e-119)
		tmp = (t / l) * sqrt(x);
	else
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 3.5e-119], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 + N[(N[(l * l), $MachinePrecision] / N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e-119

   1. Initial program 28.3%

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

   2. Taylor expanded in l around inf 3.4%

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

   3. Taylor expanded in x around inf 19.5%

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

   4. Taylor expanded in t around 0 18.0%

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

   if 3.5e-119 < t

   1. Initial program 49.2%

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

   2. Taylor expanded in x around inf 61.1%

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

      1. fma-def 61.1%

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

      2. +-commutative 61.1%

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

      3. unpow2 61.1%

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

      4. unpow2 61.1%

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

      5. fma-udef 61.1%

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

      6. mul-1-neg 61.1%

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

      7. +-commutative 61.1%

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

      8. unpow2 61.1%

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

      9. unpow2 61.1%

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

      10. fma-udef 61.1%

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

      11. *-commutative 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in x around inf 89.5%

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

      1. mul-1-neg 89.5%

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

      2. unsub-neg 89.5%

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

   7. Simplified 89.5%

      \[\leadsto \color{blue}{1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \end{array} \]

Alternative 16: 37.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t 2.05e-163)
   (/ 2.0 (* (/ l t) (/ (/ l x) t)))
   (- 1.0 (/ (+ 1.0 (/ (* l l) (* 2.0 (* t t)))) x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= 2.05e-163) {
		tmp = 2.0 / ((l / t) * ((l / x) / t));
	} else {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / 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 <= 2.05d-163) then
        tmp = 2.0d0 / ((l / t) * ((l / x) / t))
    else
        tmp = 1.0d0 - ((1.0d0 + ((l * l) / (2.0d0 * (t * t)))) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= 2.05e-163) {
		tmp = 2.0 / ((l / t) * ((l / x) / t));
	} else {
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= 2.05e-163:
		tmp = 2.0 / ((l / t) * ((l / x) / t))
	else:
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= 2.05e-163)
		tmp = Float64(2.0 / Float64(Float64(l / t) * Float64(Float64(l / x) / t)));
	else
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(l * l) / Float64(2.0 * Float64(t * t)))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 2.05e-163)
		tmp = 2.0 / ((l / t) * ((l / x) / t));
	else
		tmp = 1.0 - ((1.0 + ((l * l) / (2.0 * (t * t)))) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, 2.05e-163], N[(2.0 / N[(N[(l / t), $MachinePrecision] * N[(N[(l / x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(1.0 + N[(N[(l * l), $MachinePrecision] / N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.04999999999999991e-163

   1. Initial program 28.5%

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

   2. Taylor expanded in x around inf 12.3%

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

      1. fma-def 12.3%

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

      2. +-commutative 12.3%

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

      3. unpow2 12.3%

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

      4. unpow2 12.3%

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

      5. fma-udef 12.3%

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

      6. mul-1-neg 12.3%

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

      7. +-commutative 12.3%

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

      8. unpow2 12.3%

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

      9. unpow2 12.3%

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

      10. fma-udef 12.3%

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

      11. *-commutative 12.3%

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

   4. Simplified 12.3%

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

   5. Taylor expanded in t around 0 9.2%

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

      1. associate-/l* 9.2%

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

      2. unpow2 9.2%

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

      3. rem-square-sqrt 9.2%

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

      4. *-commutative 9.2%

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

      5. associate-/r* 8.9%

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

      6. unpow2 8.9%

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

      7. unpow2 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in l around 0 9.2%

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

      1. associate-/l* 8.9%

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

      2. unpow2 8.9%

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

      3. unpow2 8.9%

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

      4. associate-*r/ 9.0%

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

   10. Simplified 9.0%

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

       1. clear-num 9.0%

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

       2. add-cbrt-cube 9.0%

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

       3. associate-*r* 9.0%

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

       4. div-inv 9.0%

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

       5. associate-*r* 9.0%

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

       6. add-cbrt-cube 9.0%

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

       7. times-frac 9.4%

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

   12. Applied egg-rr 9.4%

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

   if 2.04999999999999991e-163 < t

   1. Initial program 47.7%

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

   2. Taylor expanded in x around inf 60.2%

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

      1. fma-def 60.2%

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

      2. +-commutative 60.2%

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

      3. unpow2 60.2%

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

      4. unpow2 60.2%

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

      5. fma-udef 60.2%

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

      6. mul-1-neg 60.2%

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

      7. +-commutative 60.2%

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

      8. unpow2 60.2%

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

      9. unpow2 60.2%

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

      10. fma-udef 60.2%

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

      11. *-commutative 60.2%

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

   4. Simplified 60.2%

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

   5. Taylor expanded in x around inf 85.7%

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

      1. mul-1-neg 85.7%

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

      2. unsub-neg 85.7%

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

   7. Simplified 85.7%

      \[\leadsto \color{blue}{1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1 + \frac{\ell \cdot \ell}{2 \cdot \left(t \cdot t\right)}}{x}\\ \end{array} \]

Alternative 17: 6.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{t \cdot t}{\ell \cdot \frac{\ell}{x}} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (* 2.0 (/ (* t t) (* l (/ l x)))))

C

double code(double x, double l, double t) {
	return 2.0 * ((t * t) / (l * (l / x)));
}

Fortran

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

Java

public static double code(double x, double l, double t) {
	return 2.0 * ((t * t) / (l * (l / x)));
}

Python

def code(x, l, t):
	return 2.0 * ((t * t) / (l * (l / x)))

Julia

function code(x, l, t)
	return Float64(2.0 * Float64(Float64(t * t) / Float64(l * Float64(l / x))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = 2.0 * ((t * t) / (l * (l / x)));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around inf 33.1%

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

   1. fma-def 33.1%

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

   2. +-commutative 33.1%

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

   3. unpow2 33.1%

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

   4. unpow2 33.1%

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

   5. fma-udef 33.1%

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

   6. mul-1-neg 33.1%

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

   7. +-commutative 33.1%

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

   8. unpow2 33.1%

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

   9. unpow2 33.1%

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

   10. fma-udef 33.1%

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

   11. *-commutative 33.1%

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

4. Simplified 33.1%

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

5. Taylor expanded in t around 0 6.3%

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

   1. associate-/l* 6.3%

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

   2. unpow2 6.3%

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

   3. rem-square-sqrt 6.3%

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

   4. *-commutative 6.3%

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

   5. associate-/r* 6.2%

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

   6. unpow2 6.2%

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

   7. unpow2 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in l around 0 6.3%

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

   1. associate-/l* 6.2%

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

   2. unpow2 6.2%

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

   3. unpow2 6.2%

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

   4. associate-*r/ 6.3%

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

10. Simplified 6.3%

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

11. Final simplification 6.3%

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

Alternative 18: 6.6% accurate, 20.5× speedup

Math

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

FPCore

(FPCore (x l t) :precision binary64 (* 2.0 (/ (* x (* t t)) (* l l))))

C

double code(double x, double l, double t) {
	return 2.0 * ((x * (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 = 2.0d0 * ((x * (t * t)) / (l * l))
end function

Java

public static double code(double x, double l, double t) {
	return 2.0 * ((x * (t * t)) / (l * l));
}

Python

def code(x, l, t):
	return 2.0 * ((x * (t * t)) / (l * l))

Julia

function code(x, l, t)
	return Float64(2.0 * Float64(Float64(x * Float64(t * t)) / Float64(l * l)))
end

MATLAB

function tmp = code(x, l, t)
	tmp = 2.0 * ((x * (t * t)) / (l * l));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around inf 33.1%

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

   1. fma-def 33.1%

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

   2. +-commutative 33.1%

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

   3. unpow2 33.1%

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

   4. unpow2 33.1%

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

   5. fma-udef 33.1%

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

   6. mul-1-neg 33.1%

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

   7. +-commutative 33.1%

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

   8. unpow2 33.1%

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

   9. unpow2 33.1%

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

   10. fma-udef 33.1%

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

   11. *-commutative 33.1%

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

4. Simplified 33.1%

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

5. Taylor expanded in t around 0 6.3%

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

   1. associate-/l* 6.3%

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

   2. unpow2 6.3%

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

   3. rem-square-sqrt 6.3%

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

   4. *-commutative 6.3%

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

   5. associate-/r* 6.2%

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

   6. unpow2 6.2%

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

   7. unpow2 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in l around 0 6.3%

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

   1. associate-/l* 6.2%

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

   2. unpow2 6.2%

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

   3. unpow2 6.2%

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

   4. associate-*r/ 6.3%

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

10. Simplified 6.3%

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

11. Taylor expanded in t around 0 6.3%

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

    1. unpow2 6.3%

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

    2. *-commutative 6.3%

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

    3. unpow2 6.3%

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

13. Simplified 6.3%

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

14. Final simplification 6.3%

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

Alternative 19: 7.2% accurate, 20.5× speedup

Math

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

FPCore

(FPCore (x l t) :precision binary64 (* 2.0 (* (/ t l) (/ t (/ l x)))))

C

double code(double x, double l, double t) {
	return 2.0 * ((t / l) * (t / (l / x)));
}

Fortran

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

Java

public static double code(double x, double l, double t) {
	return 2.0 * ((t / l) * (t / (l / x)));
}

Python

def code(x, l, t):
	return 2.0 * ((t / l) * (t / (l / x)))

Julia

function code(x, l, t)
	return Float64(2.0 * Float64(Float64(t / l) * Float64(t / Float64(l / x))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = 2.0 * ((t / l) * (t / (l / x)));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around inf 33.1%

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

   1. fma-def 33.1%

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

   2. +-commutative 33.1%

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

   3. unpow2 33.1%

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

   4. unpow2 33.1%

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

   5. fma-udef 33.1%

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

   6. mul-1-neg 33.1%

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

   7. +-commutative 33.1%

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

   8. unpow2 33.1%

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

   9. unpow2 33.1%

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

   10. fma-udef 33.1%

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

   11. *-commutative 33.1%

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

4. Simplified 33.1%

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

5. Taylor expanded in t around 0 6.3%

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

   1. associate-/l* 6.3%

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

   2. unpow2 6.3%

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

   3. rem-square-sqrt 6.3%

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

   4. *-commutative 6.3%

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

   5. associate-/r* 6.2%

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

   6. unpow2 6.2%

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

   7. unpow2 6.2%

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

7. Simplified 6.2%

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

   1. add-cbrt-cube 6.1%

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

   2. associate-/l* 6.1%

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

   3. associate-/r/ 6.1%

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

   4. associate-/l* 6.1%

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

   5. associate-/r/ 6.1%

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

   6. associate-/l* 6.2%

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

   7. associate-/r/ 6.2%

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

9. Applied egg-rr 6.2%

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

    1. associate-*l* 6.2%

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

    2. *-commutative 6.2%

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

    3. *-commutative 6.2%

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

    4. *-commutative 6.2%

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

11. Simplified 6.2%

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

    1. div-inv 6.2%

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

    2. associate-*r* 6.2%

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

    3. add-cbrt-cube 6.3%

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

    4. clear-num 6.3%

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

    5. *-commutative 6.3%

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

    6. times-frac 6.6%

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

13. Applied egg-rr 6.6%

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

14. Final simplification 6.6%

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

Alternative 20: 7.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t}} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (/ 2.0 (* (/ l t) (/ (/ l x) t))))

C

double code(double x, double l, double t) {
	return 2.0 / ((l / t) * ((l / x) / t));
}

Fortran

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

Java

public static double code(double x, double l, double t) {
	return 2.0 / ((l / t) * ((l / x) / t));
}

Python

def code(x, l, t):
	return 2.0 / ((l / t) * ((l / x) / t))

Julia

function code(x, l, t)
	return Float64(2.0 / Float64(Float64(l / t) * Float64(Float64(l / x) / t)))
end

MATLAB

function tmp = code(x, l, t)
	tmp = 2.0 / ((l / t) * ((l / x) / t));
end

Wolfram

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

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in x around inf 33.1%

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

   1. fma-def 33.1%

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

   2. +-commutative 33.1%

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

   3. unpow2 33.1%

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

   4. unpow2 33.1%

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

   5. fma-udef 33.1%

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

   6. mul-1-neg 33.1%

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

   7. +-commutative 33.1%

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

   8. unpow2 33.1%

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

   9. unpow2 33.1%

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

   10. fma-udef 33.1%

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

   11. *-commutative 33.1%

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

4. Simplified 33.1%

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

5. Taylor expanded in t around 0 6.3%

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

   1. associate-/l* 6.3%

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

   2. unpow2 6.3%

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

   3. rem-square-sqrt 6.3%

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

   4. *-commutative 6.3%

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

   5. associate-/r* 6.2%

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

   6. unpow2 6.2%

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

   7. unpow2 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in l around 0 6.3%

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

   1. associate-/l* 6.2%

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

   2. unpow2 6.2%

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

   3. unpow2 6.2%

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

   4. associate-*r/ 6.3%

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

10. Simplified 6.3%

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

    1. clear-num 6.3%

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

    2. add-cbrt-cube 6.2%

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

    3. associate-*r* 6.2%

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

    4. div-inv 6.2%

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

    5. associate-*r* 6.2%

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

    6. add-cbrt-cube 6.3%

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

    7. times-frac 6.6%

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

12. Applied egg-rr 6.6%

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

13. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2023187 
(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)))))