Toniolo and Linder, Equation (7)

Percentage Accurate: 33.7% → 83.6%

Time: 21.6s

Alternatives: 9

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.7% 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: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-236}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{l\_m}\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{-217}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t\_m \leq 1.18 \cdot 10^{-206}:\\ \;\;\;\;t\_m \cdot \frac{t\_3}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{-172}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 0.0053:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (sqrt (* 2.0 (fma x 0.5 -0.5)))))
   (*
    t_s
    (if (<= t_m 2.6e-236)
      (/ (* t_m t_3) l_m)
      (if (<= t_m 4.5e-217)
        (+ 1.0 (/ (+ (/ 0.5 x) -1.0) x))
        (if (<= t_m 1.18e-206)
          (* t_m (/ t_3 l_m))
          (if (<= t_m 1.9e-172)
            (+ 1.0 (/ -1.0 x))
            (if (<= t_m 0.0053)
              (*
               (sqrt 2.0)
               (/
                t_m
                (sqrt
                 (+
                  (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
                  (/ (+ t_2 (pow l_m 2.0)) x)))))
              (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = sqrt((2.0 * fma(x, 0.5, -0.5)));
	double tmp;
	if (t_m <= 2.6e-236) {
		tmp = (t_m * t_3) / l_m;
	} else if (t_m <= 4.5e-217) {
		tmp = 1.0 + (((0.5 / x) + -1.0) / x);
	} else if (t_m <= 1.18e-206) {
		tmp = t_m * (t_3 / l_m);
	} else if (t_m <= 1.9e-172) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 0.0053) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = sqrt(Float64(2.0 * fma(x, 0.5, -0.5)))
	tmp = 0.0
	if (t_m <= 2.6e-236)
		tmp = Float64(Float64(t_m * t_3) / l_m);
	elseif (t_m <= 4.5e-217)
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 / x) + -1.0) / x));
	elseif (t_m <= 1.18e-206)
		tmp = Float64(t_m * Float64(t_3 / l_m));
	elseif (t_m <= 1.9e-172)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 0.0053)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.6e-236], N[(N[(t$95$m * t$95$3), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 4.5e-217], N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.18e-206], N[(t$95$m * N[(t$95$3 / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e-172], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.0053], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 6 regimes

2. if t < 2.6e-236

   1. Initial program 36.0%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in l around inf 3.2%

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

      1. *-commutative 3.2%

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

      2. associate--l+ 10.9%

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

      3. sub-neg 10.9%

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

      4. metadata-eval 10.9%

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

      5. +-commutative 10.9%

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

      6. sub-neg 10.9%

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

      7. metadata-eval 10.9%

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

      8. +-commutative 10.9%

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

   7. Simplified 10.9%

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

   8. Taylor expanded in x around 0 17.9%

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

      1. associate-*r* 18.0%

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

      2. clear-num 17.7%

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

      3. un-div-inv 17.6%

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

      4. sqrt-unprod 17.7%

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

      5. *-commutative 17.7%

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

      6. fma-neg 17.7%

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

      7. metadata-eval 17.7%

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

   10. Applied egg-rr 17.7%

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

       1. associate-/r/ 19.4%

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

       2. associate-*l/ 19.3%

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

       3. associate-*r/ 18.0%

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

       4. *-commutative 18.0%

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

   12. Simplified 18.0%

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

       1. associate-*l/ 19.3%

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

   14. Applied egg-rr 19.3%

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

   if 2.6e-236 < t < 4.4999999999999999e-217

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

   6. Taylor expanded in t around 0 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. associate-/r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{\frac{0.5}{x} + -1}{x}} \]

   if 4.4999999999999999e-217 < t < 1.17999999999999994e-206

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-neg 100.0%

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

      7. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

   12. Simplified 100.0%

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

   if 1.17999999999999994e-206 < t < 1.89999999999999993e-172

   1. Initial program 2.5%

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

   3. Simplified 2.5%

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

   5. Taylor expanded in l around 0 80.6%

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

   6. Taylor expanded in x around inf 80.6%

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

   if 1.89999999999999993e-172 < t < 0.00530000000000000002

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in x around inf 90.3%

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

   if 0.00530000000000000002 < t

   1. Initial program 37.7%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in l around 0 91.7%

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

   6. Taylor expanded in t around 0 91.9%

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

      1. clear-num 91.9%

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

      2. sub-neg 91.9%

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

      3. metadata-eval 91.9%

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

      4. sqrt-div 91.9%

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

      5. metadata-eval 91.9%

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

      6. +-commutative 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. pow1/2 91.9%

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

      2. +-commutative 91.9%

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

      3. pow-flip 91.9%

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

      4. metadata-eval 91.9%

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

   10. Applied egg-rr 91.9%

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

       1. +-commutative 91.9%

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

       2. +-commutative 91.9%

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

   12. Simplified 91.9%

       \[\leadsto \color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-217}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-206}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-172}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 0.0053:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x + 1}{x + -1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{t\_2 \cdot \left(l\_m \cdot l\_m + 2 \cdot \left(t\_m \cdot t\_m\right)\right) - l\_m \cdot l\_m}} \leq 2:\\ \;\;\;\;{t\_2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ x 1.0) (+ x -1.0))))
   (*
    t_s
    (if (<=
         (/
          (* t_m (sqrt 2.0))
          (sqrt (- (* t_2 (+ (* l_m l_m) (* 2.0 (* t_m t_m)))) (* l_m l_m))))
         2.0)
      (pow t_2 -0.5)
      (* t_m (/ (sqrt (* 2.0 (fma x 0.5 -0.5))) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (((t_m * sqrt(2.0)) / sqrt(((t_2 * ((l_m * l_m) + (2.0 * (t_m * t_m)))) - (l_m * l_m)))) <= 2.0) {
		tmp = pow(t_2, -0.5);
	} else {
		tmp = t_m * (sqrt((2.0 * fma(x, 0.5, -0.5))) / l_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(Float64(t_2 * Float64(Float64(l_m * l_m) + Float64(2.0 * Float64(t_m * t_m)))) - Float64(l_m * l_m)))) <= 2.0)
		tmp = t_2 ^ -0.5;
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) / l_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 * N[(N[(l$95$m * l$95$m), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[Power[t$95$2, -0.5], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 53.9%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in l around 0 43.9%

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

   6. Taylor expanded in t around 0 44.0%

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

      1. clear-num 44.0%

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

      2. sub-neg 44.0%

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

      3. metadata-eval 44.0%

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

      4. sqrt-div 44.0%

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

      5. metadata-eval 44.0%

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

      6. +-commutative 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. pow1/2 44.0%

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

      2. +-commutative 44.0%

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

      3. pow-flip 44.0%

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

      4. metadata-eval 44.0%

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

   10. Applied egg-rr 44.0%

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

       1. +-commutative 44.0%

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

       2. +-commutative 44.0%

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

   12. Simplified 44.0%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   5. Taylor expanded in l around inf 3.8%

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

      1. *-commutative 3.8%

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

      2. associate--l+ 21.3%

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

      3. sub-neg 21.3%

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

      4. metadata-eval 21.3%

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

      5. +-commutative 21.3%

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

      6. sub-neg 21.3%

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

      7. metadata-eval 21.3%

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

      8. +-commutative 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in x around 0 33.3%

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

      1. associate-*r* 33.4%

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

      2. clear-num 31.4%

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

      3. un-div-inv 31.4%

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

      4. sqrt-unprod 31.4%

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

      5. *-commutative 31.4%

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

      6. fma-neg 31.4%

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

      7. metadata-eval 31.4%

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

   10. Applied egg-rr 31.4%

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

       1. associate-/r/ 35.1%

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

   12. Simplified 35.1%

       \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 2:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t\_m}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.3 \cdot 10^{-237}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{-215}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{-207}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt (* 2.0 (fma x 0.5 -0.5))) (/ t_m l_m))))
   (*
    t_s
    (if (<= t_m 9.3e-237)
      t_2
      (if (<= t_m 3.2e-215)
        (+ 1.0 (/ (+ (/ 0.5 x) -1.0) x))
        (if (<= t_m 6.2e-207)
          t_2
          (if (<= t_m 1.9e-122)
            (+ 1.0 (/ -1.0 x))
            (if (<= t_m 6.4e-115)
              t_2
              (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt((2.0 * fma(x, 0.5, -0.5))) * (t_m / l_m);
	double tmp;
	if (t_m <= 9.3e-237) {
		tmp = t_2;
	} else if (t_m <= 3.2e-215) {
		tmp = 1.0 + (((0.5 / x) + -1.0) / x);
	} else if (t_m <= 6.2e-207) {
		tmp = t_2;
	} else if (t_m <= 1.9e-122) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 6.4e-115) {
		tmp = t_2;
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) * Float64(t_m / l_m))
	tmp = 0.0
	if (t_m <= 9.3e-237)
		tmp = t_2;
	elseif (t_m <= 3.2e-215)
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 / x) + -1.0) / x));
	elseif (t_m <= 6.2e-207)
		tmp = t_2;
	elseif (t_m <= 1.9e-122)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 6.4e-115)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.3e-237], t$95$2, If[LessEqual[t$95$m, 3.2e-215], N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e-207], t$95$2, If[LessEqual[t$95$m, 1.9e-122], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e-115], t$95$2, N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 9.2999999999999995e-237 or 3.2000000000000001e-215 < t < 6.2000000000000003e-207 or 1.9e-122 < t < 6.4e-115

   1. Initial program 34.9%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 12.9%

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

      3. sub-neg 12.9%

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

      4. metadata-eval 12.9%

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

      5. +-commutative 12.9%

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

      6. sub-neg 12.9%

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

      7. metadata-eval 12.9%

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

      8. +-commutative 12.9%

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

   7. Simplified 12.9%

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

   8. Taylor expanded in x around 0 19.7%

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

      1. associate-*r* 19.7%

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

      2. clear-num 19.4%

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

      3. un-div-inv 19.4%

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

      4. sqrt-unprod 19.4%

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

      5. *-commutative 19.4%

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

      6. fma-neg 19.4%

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

      7. metadata-eval 19.4%

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

   10. Applied egg-rr 19.4%

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

       1. associate-/r/ 21.1%

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

       2. associate-*l/ 21.0%

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

       3. associate-*r/ 19.8%

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

       4. *-commutative 19.8%

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

   12. Simplified 19.8%

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

   if 9.2999999999999995e-237 < t < 3.2000000000000001e-215

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

   6. Taylor expanded in t around 0 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. associate-/r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{\frac{0.5}{x} + -1}{x}} \]

   if 6.2000000000000003e-207 < t < 1.9e-122

   1. Initial program 24.6%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in l around 0 63.1%

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

   6. Taylor expanded in x around inf 63.2%

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

   if 6.4e-115 < t

   1. Initial program 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in l around 0 85.9%

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

   6. Taylor expanded in t around 0 86.1%

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

      1. clear-num 86.1%

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

      2. sub-neg 86.1%

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

      3. metadata-eval 86.1%

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

      4. sqrt-div 86.1%

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

      5. metadata-eval 86.1%

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

      6. +-commutative 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. pow1/2 86.1%

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

      2. +-commutative 86.1%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. +-commutative 86.1%

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

       2. +-commutative 86.1%

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

   12. Simplified 86.1%

       \[\leadsto \color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-215}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot \left(\frac{t\_m}{l\_m} \cdot \sqrt{x \cdot 0.5}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-236}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{-215}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t\_m \leq 6.1 \cdot 10^{-207}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{-125}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) (* (/ t_m l_m) (sqrt (* x 0.5))))))
   (*
    t_s
    (if (<= t_m 2.7e-236)
      t_2
      (if (<= t_m 8.6e-215)
        (+ 1.0 (/ (+ (/ 0.5 x) -1.0) x))
        (if (<= t_m 6.1e-207)
          t_2
          (if (<= t_m 5.4e-125)
            (+ 1.0 (/ -1.0 x))
            (if (<= t_m 6.4e-115)
              t_2
              (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = sqrt(2.0) * ((t_m / l_m) * sqrt((x * 0.5)));
	double tmp;
	if (t_m <= 2.7e-236) {
		tmp = t_2;
	} else if (t_m <= 8.6e-215) {
		tmp = 1.0 + (((0.5 / x) + -1.0) / x);
	} else if (t_m <= 6.1e-207) {
		tmp = t_2;
	} else if (t_m <= 5.4e-125) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 6.4e-115) {
		tmp = t_2;
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * ((t_m / l_m) * sqrt((x * 0.5d0)))
    if (t_m <= 2.7d-236) then
        tmp = t_2
    else if (t_m <= 8.6d-215) then
        tmp = 1.0d0 + (((0.5d0 / x) + (-1.0d0)) / x)
    else if (t_m <= 6.1d-207) then
        tmp = t_2
    else if (t_m <= 5.4d-125) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 6.4d-115) then
        tmp = t_2
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * ((t_m / l_m) * Math.sqrt((x * 0.5)));
	double tmp;
	if (t_m <= 2.7e-236) {
		tmp = t_2;
	} else if (t_m <= 8.6e-215) {
		tmp = 1.0 + (((0.5 / x) + -1.0) / x);
	} else if (t_m <= 6.1e-207) {
		tmp = t_2;
	} else if (t_m <= 5.4e-125) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 6.4e-115) {
		tmp = t_2;
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * ((t_m / l_m) * math.sqrt((x * 0.5)))
	tmp = 0
	if t_m <= 2.7e-236:
		tmp = t_2
	elif t_m <= 8.6e-215:
		tmp = 1.0 + (((0.5 / x) + -1.0) / x)
	elif t_m <= 6.1e-207:
		tmp = t_2
	elif t_m <= 5.4e-125:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 6.4e-115:
		tmp = t_2
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * Float64(Float64(t_m / l_m) * sqrt(Float64(x * 0.5))))
	tmp = 0.0
	if (t_m <= 2.7e-236)
		tmp = t_2;
	elseif (t_m <= 8.6e-215)
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 / x) + -1.0) / x));
	elseif (t_m <= 6.1e-207)
		tmp = t_2;
	elseif (t_m <= 5.4e-125)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 6.4e-115)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * ((t_m / l_m) * sqrt((x * 0.5)));
	tmp = 0.0;
	if (t_m <= 2.7e-236)
		tmp = t_2;
	elseif (t_m <= 8.6e-215)
		tmp = 1.0 + (((0.5 / x) + -1.0) / x);
	elseif (t_m <= 6.1e-207)
		tmp = t_2;
	elseif (t_m <= 5.4e-125)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 6.4e-115)
		tmp = t_2;
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-236], t$95$2, If[LessEqual[t$95$m, 8.6e-215], N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.1e-207], t$95$2, If[LessEqual[t$95$m, 5.4e-125], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.4e-115], t$95$2, N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.7e-236 or 8.60000000000000049e-215 < t < 6.10000000000000008e-207 or 5.3999999999999995e-125 < t < 6.4e-115

   1. Initial program 34.9%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

      2. associate--l+ 12.9%

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

      3. sub-neg 12.9%

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

      4. metadata-eval 12.9%

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

      5. +-commutative 12.9%

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

      6. sub-neg 12.9%

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

      7. metadata-eval 12.9%

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

      8. +-commutative 12.9%

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

   7. Simplified 12.9%

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

   8. Taylor expanded in x around inf 19.7%

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

      1. *-commutative 19.7%

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

   10. Simplified 19.7%

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

   if 2.7e-236 < t < 8.60000000000000049e-215

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

   6. Taylor expanded in t around 0 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. associate-/r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. rem-square-sqrt 0.0%

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

      8. unpow2 0.0%

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

      9. associate-*l/ 0.0%

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

      10. *-commutative 0.0%

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

      11. div-sub 0.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{\frac{0.5}{x} + -1}{x}} \]

   if 6.10000000000000008e-207 < t < 5.3999999999999995e-125

   1. Initial program 24.6%

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

   3. Simplified 24.4%

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

   5. Taylor expanded in l around 0 63.1%

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

   6. Taylor expanded in x around inf 63.2%

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

   if 6.4e-115 < t

   1. Initial program 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in l around 0 85.9%

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

   6. Taylor expanded in t around 0 86.1%

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

      1. clear-num 86.1%

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

      2. sub-neg 86.1%

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

      3. metadata-eval 86.1%

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

      4. sqrt-div 86.1%

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

      5. metadata-eval 86.1%

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

      6. +-commutative 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. pow1/2 86.1%

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

      2. +-commutative 86.1%

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

      3. pow-flip 86.1%

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

      4. metadata-eval 86.1%

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

   10. Applied egg-rr 86.1%

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

       1. +-commutative 86.1%

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

       2. +-commutative 86.1%

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

   12. Simplified 86.1%

       \[\leadsto \color{blue}{{\left(\frac{x + 1}{-1 + x}\right)}^{-0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{x \cdot 0.5}\right)\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-215}:\\ \;\;\;\;1 + \frac{\frac{0.5}{x} + -1}{x}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{x \cdot 0.5}\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-125}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{x \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot {\left(\frac{x + 1}{x + -1}\right)}^{-0.5} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * pow(((x + 1.0) / (x + -1.0)), -0.5);
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.pow(((x + 1.0) / (x + -1.0)), -0.5)

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * (Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (((x + 1.0) / (x + -1.0)) ^ -0.5);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 36.5%

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

5. Taylor expanded in l around 0 42.4%

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

6. Taylor expanded in t around 0 42.4%

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

   1. clear-num 42.4%

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

   2. sub-neg 42.4%

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

   3. metadata-eval 42.4%

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

   4. sqrt-div 42.4%

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

   5. metadata-eval 42.4%

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

   6. +-commutative 42.4%

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

8. Applied egg-rr 42.4%

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

   1. pow1/2 42.4%

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

   2. +-commutative 42.4%

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

   3. pow-flip 42.5%

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

   4. metadata-eval 42.5%

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

10. Applied egg-rr 42.5%

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

    1. +-commutative 42.5%

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

    2. +-commutative 42.5%

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

12. Simplified 42.5%

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

13. Final simplification 42.5%

    \[\leadsto {\left(\frac{x + 1}{x + -1}\right)}^{-0.5} \]
14. Add Preprocessing

Alternative 6: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x + -1}{x + 1}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ x -1.0) (+ x 1.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * sqrt(((x + -1.0) / (x + 1.0)));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.sqrt(((x + -1.0) / (x + 1.0)));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.sqrt(((x + -1.0) / (x + 1.0)))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * sqrt(((x + -1.0) / (x + 1.0)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 36.5%

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

5. Taylor expanded in l around 0 42.4%

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

6. Taylor expanded in t around 0 42.4%

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

7. Final simplification 42.4%

   \[\leadsto \sqrt{\frac{x + -1}{x + 1}} \]
8. Add Preprocessing

Alternative 7: 76.6% accurate, 25.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{\frac{0.5}{x} + -1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (+ (/ 0.5 x) -1.0) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (((0.5 / x) + -1.0) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((0.5d0 / x) + (-1.0d0)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (((0.5 / x) + -1.0) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (((0.5 / x) + -1.0) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(Float64(0.5 / x) + -1.0) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (((0.5 / x) + -1.0) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 36.5%

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

5. Taylor expanded in l around 0 42.4%

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

6. Taylor expanded in t around 0 42.4%

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

7. Taylor expanded in x around inf 42.2%

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

   1. associate--l+ 42.2%

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

   2. unpow2 42.2%

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

   3. associate-/r* 42.2%

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

   4. metadata-eval 42.2%

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

   5. metadata-eval 42.2%

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

   6. metadata-eval 42.2%

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

   7. rem-square-sqrt 0.0%

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

   8. unpow2 0.0%

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

   9. associate-*l/ 0.0%

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

   10. *-commutative 0.0%

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

   11. div-sub 0.0%

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

9. Simplified 42.2%

   \[\leadsto \color{blue}{1 + \frac{\frac{0.5}{x} + -1}{x}} \]
10. Add Preprocessing

Alternative 8: 76.4% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 36.5%

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

5. Taylor expanded in l around 0 42.4%

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

6. Taylor expanded in x around inf 42.1%

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

7. Final simplification 42.1%

   \[\leadsto 1 + \frac{-1}{x} \]
8. Add Preprocessing

Alternative 9: 75.7% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 36.5%

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

5. Taylor expanded in l around 0 42.4%

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

6. Taylor expanded in x around inf 41.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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)))))