Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 78.7%

Time: 22.1s

Alternatives: 10

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: 32.9% 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: 78.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 + x}{x + 1}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 31500:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;l_m \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{t_2}, 2 \cdot \frac{{l_m}^{2}}{x}\right)}}\\ \mathbf{elif}\;l_m \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ -1.0 x) (+ x 1.0))))
   (*
    t_s
    (if (<= l_m 31500.0)
      (+ 1.0 (/ -1.0 x))
      (if (<= l_m 1.7e+124)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt (fma 2.0 (/ (pow t_m 2.0) t_2) (* 2.0 (/ (pow l_m 2.0) x))))))
        (if (<= l_m 8.5e+209)
          (sqrt t_2)
          (* 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 = (-1.0 + x) / (x + 1.0);
	double tmp;
	if (l_m <= 31500.0) {
		tmp = 1.0 + (-1.0 / x);
	} else if (l_m <= 1.7e+124) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) / t_2), (2.0 * (pow(l_m, 2.0) / x)))));
	} else if (l_m <= 8.5e+209) {
		tmp = sqrt(t_2);
	} 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(-1.0 + x) / Float64(x + 1.0))
	tmp = 0.0
	if (l_m <= 31500.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (l_m <= 1.7e+124)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) / t_2), Float64(2.0 * Float64((l_m ^ 2.0) / x))))));
	elseif (l_m <= 8.5e+209)
		tmp = sqrt(t_2);
	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[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 31500.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 1.7e+124], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] + N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 8.5e+209], N[Sqrt[t$95$2], $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 4 regimes

2. if l < 31500

   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%

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

   5. Taylor expanded in t around inf 48.5%

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

   6. Taylor expanded in x around inf 48.7%

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

   if 31500 < l < 1.7e124

   1. Initial program 21.5%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in l around 0 23.8%

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

      1. fma-def 23.8%

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

      2. associate-/l* 44.6%

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

      3. sub-neg 44.6%

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

      4. metadata-eval 44.6%

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

      5. +-commutative 44.6%

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

      6. +-commutative 44.6%

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

      7. associate--l+ 53.9%

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

      8. sub-neg 53.9%

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

      9. metadata-eval 53.9%

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

      10. +-commutative 53.9%

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

      11. sub-neg 53.9%

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

      12. metadata-eval 53.9%

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

      13. +-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in x around inf 82.5%

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

   if 1.7e124 < l < 8.50000000000000062e209

   1. Initial program 14.5%

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

   3. Simplified 14.5%

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

   5. Taylor expanded in t around inf 47.7%

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

   6. Taylor expanded in t around 0 48.1%

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

   if 8.50000000000000062e209 < l

   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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 52.4%

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

      3. sub-neg 52.4%

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

      4. metadata-eval 52.4%

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

      5. +-commutative 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in x around 0 81.9%

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

      1. clear-num 81.9%

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

      2. un-div-inv 81.9%

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

      3. fma-neg 81.9%

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

      4. metadata-eval 81.9%

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

   10. Applied egg-rr 81.9%

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

       1. associate-*r/ 81.9%

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

       2. frac-2neg 81.9%

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

       3. metadata-eval 81.9%

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

       4. fma-neg 81.9%

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

       5. sqrt-unprod 81.9%

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

       6. fma-neg 81.9%

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

       7. metadata-eval 81.9%

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

       8. distribute-neg-frac 81.9%

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

   12. Applied egg-rr 81.9%

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

       1. associate-/r/ 81.9%

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

       2. neg-mul-1 81.9%

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

       3. neg-mul-1 81.9%

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

       4. times-frac 81.9%

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

       5. metadata-eval 81.9%

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

       6. fma-udef 81.9%

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

       7. *-commutative 81.9%

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

       8. fma-def 81.9%

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

   14. Simplified 81.9%

       \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\right) \cdot t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 31500:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{0.5 \cdot \frac{2 \cdot \left({t_m}^{2} + {t_m}^{2}\right) + 2 \cdot {l_m}^{2}}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{{l_m}^{2} + t_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 8.8e-165)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (*
          0.5
          (/
           (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l_m 2.0)))
           (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.3e+92)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
            (/ (+ (pow l_m 2.0) t_2) x)))))
        (sqrt (/ (+ -1.0 x) (+ 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) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 8.8e-165) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l_m, 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.3e+92) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((pow(l_m, 2.0) + t_2) / x))));
	} else {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 8.8d-165) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * (((2.0d0 * ((t_m ** 2.0d0) + (t_m ** 2.0d0))) + (2.0d0 * (l_m ** 2.0d0))) / (t_m * (sqrt(2.0d0) * x)))) + (t_m * sqrt(2.0d0))))
    else if (t_m <= 2.3d+92) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + (((l_m ** 2.0d0) + t_2) / x))))
    else
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    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 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 8.8e-165) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (Math.pow(t_m, 2.0) + Math.pow(t_m, 2.0))) + (2.0 * Math.pow(l_m, 2.0))) / (t_m * (Math.sqrt(2.0) * x)))) + (t_m * Math.sqrt(2.0))));
	} else if (t_m <= 2.3e+92) {
		tmp = t_m * (Math.sqrt(2.0) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((Math.pow(l_m, 2.0) + t_2) / x))));
	} else {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	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 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 8.8e-165:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (math.pow(t_m, 2.0) + math.pow(t_m, 2.0))) + (2.0 * math.pow(l_m, 2.0))) / (t_m * (math.sqrt(2.0) * x)))) + (t_m * math.sqrt(2.0))))
	elif t_m <= 2.3e+92:
		tmp = t_m * (math.sqrt(2.0) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((math.pow(l_m, 2.0) + t_2) / x))))
	else:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	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))
	tmp = 0.0
	if (t_m <= 8.8e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l_m ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.3e+92)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64((l_m ^ 2.0) + t_2) / x)))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 8.8e-165)
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * ((t_m ^ 2.0) + (t_m ^ 2.0))) + (2.0 * (l_m ^ 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	elseif (t_m <= 2.3e+92)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + (((l_m ^ 2.0) + t_2) / x))));
	else
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	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[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.8e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+92], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / 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[(N[Power[l$95$m, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 8.7999999999999995e-165

   1. Initial program 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in l around 0 23.3%

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

      1. fma-def 23.3%

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

      2. associate-/l* 33.1%

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

      3. sub-neg 33.1%

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

      4. metadata-eval 33.1%

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

      5. +-commutative 33.1%

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

      6. +-commutative 33.1%

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

      7. associate--l+ 44.3%

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

      8. sub-neg 44.3%

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

      9. metadata-eval 44.3%

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

      10. +-commutative 44.3%

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

      11. sub-neg 44.3%

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

      12. metadata-eval 44.3%

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

      13. +-commutative 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in x around inf 20.4%

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

   if 8.7999999999999995e-165 < t < 2.29999999999999998e92

   1. Initial program 49.7%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in x around inf 83.5%

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

   if 2.29999999999999998e92 < t

   1. Initial program 25.5%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in t around inf 98.1%

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

   6. Taylor expanded in t around 0 98.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 + x}{x + 1}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.16 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{0.5 \cdot \frac{2 \cdot \left({t_m}^{2} + {t_m}^{2}\right) + 2 \cdot {l_m}^{2}}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 2.75 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{t_2}, {l_m}^{2} \cdot \left(\frac{1}{-1 + x} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ -1.0 x) (+ x 1.0))))
   (*
    t_s
    (if (<= t_m 1.16e-164)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (*
          0.5
          (/
           (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l_m 2.0)))
           (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 2.75e+93)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (fma
            2.0
            (/ (pow t_m 2.0) t_2)
            (*
             (pow l_m 2.0)
             (+ (/ 1.0 (+ -1.0 x)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))))))
        (sqrt t_2))))))

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 = (-1.0 + x) / (x + 1.0);
	double tmp;
	if (t_m <= 1.16e-164) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l_m, 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 2.75e+93) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) / t_2), (pow(l_m, 2.0) * ((1.0 / (-1.0 + x)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))))));
	} else {
		tmp = sqrt(t_2);
	}
	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(-1.0 + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_m <= 1.16e-164)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l_m ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 2.75e+93)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) / t_2), Float64((l_m ^ 2.0) * Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0)))))))));
	else
		tmp = sqrt(t_2);
	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[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.16e-164], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.75e+93], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[t$95$2], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.16e-164

   1. Initial program 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in l around 0 23.3%

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

      1. fma-def 23.3%

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

      2. associate-/l* 33.1%

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

      3. sub-neg 33.1%

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

      4. metadata-eval 33.1%

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

      5. +-commutative 33.1%

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

      6. +-commutative 33.1%

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

      7. associate--l+ 44.3%

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

      8. sub-neg 44.3%

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

      9. metadata-eval 44.3%

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

      10. +-commutative 44.3%

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

      11. sub-neg 44.3%

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

      12. metadata-eval 44.3%

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

      13. +-commutative 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in x around inf 20.4%

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

   if 1.16e-164 < t < 2.75000000000000015e93

   1. Initial program 49.7%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in l around 0 61.3%

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

      1. fma-def 61.3%

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

      2. associate-/l* 69.5%

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

      3. sub-neg 69.5%

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

      4. metadata-eval 69.5%

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

      5. +-commutative 69.5%

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

      6. +-commutative 69.5%

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

      7. associate--l+ 75.6%

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

      8. sub-neg 75.6%

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

      9. metadata-eval 75.6%

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

      10. +-commutative 75.6%

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

      11. sub-neg 75.6%

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

      12. metadata-eval 75.6%

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

      13. +-commutative 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around inf 83.4%

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

   if 2.75000000000000015e93 < t

   1. Initial program 25.5%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in t around inf 98.1%

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

   6. Taylor expanded in t around 0 98.4%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, {\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{-1 + x}{x + 1}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{0.5 \cdot \frac{2 \cdot \left({t_m}^{2} + {t_m}^{2}\right) + 2 \cdot {l_m}^{2}}{t_m \cdot \left(\sqrt{2} \cdot x\right)} + t_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t_m \leq 1.9 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{t_2}, 2 \cdot \frac{{l_m}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (+ -1.0 x) (+ x 1.0))))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (*
       (sqrt 2.0)
       (/
        t_m
        (+
         (*
          0.5
          (/
           (+ (* 2.0 (+ (pow t_m 2.0) (pow t_m 2.0))) (* 2.0 (pow l_m 2.0)))
           (* t_m (* (sqrt 2.0) x))))
         (* t_m (sqrt 2.0)))))
      (if (<= t_m 1.9e+92)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt (fma 2.0 (/ (pow t_m 2.0) t_2) (* 2.0 (/ (pow l_m 2.0) x))))))
        (sqrt t_2))))))

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 = (-1.0 + x) / (x + 1.0);
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((2.0 * (pow(t_m, 2.0) + pow(t_m, 2.0))) + (2.0 * pow(l_m, 2.0))) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.9e+92) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) / t_2), (2.0 * (pow(l_m, 2.0) / x)))));
	} else {
		tmp = sqrt(t_2);
	}
	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(-1.0 + x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) + (t_m ^ 2.0))) + Float64(2.0 * (l_m ^ 2.0))) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.9e+92)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) / t_2), Float64(2.0 * Float64((l_m ^ 2.0) / x))))));
	else
		tmp = sqrt(t_2);
	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[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+92], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] + N[(2.0 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[t$95$2], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 26.6%

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

   3. Simplified 26.6%

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

   5. Taylor expanded in l around 0 23.3%

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

      1. fma-def 23.3%

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

      2. associate-/l* 33.1%

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

      3. sub-neg 33.1%

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

      4. metadata-eval 33.1%

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

      5. +-commutative 33.1%

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

      6. +-commutative 33.1%

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

      7. associate--l+ 44.3%

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

      8. sub-neg 44.3%

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

      9. metadata-eval 44.3%

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

      10. +-commutative 44.3%

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

      11. sub-neg 44.3%

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

      12. metadata-eval 44.3%

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

      13. +-commutative 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in x around inf 20.4%

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

   if 7.19999999999999969e-165 < t < 1.9e92

   1. Initial program 49.7%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in l around 0 61.3%

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

      1. fma-def 61.3%

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

      2. associate-/l* 69.5%

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

      3. sub-neg 69.5%

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

      4. metadata-eval 69.5%

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

      5. +-commutative 69.5%

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

      6. +-commutative 69.5%

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

      7. associate--l+ 75.6%

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

      8. sub-neg 75.6%

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

      9. metadata-eval 75.6%

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

      10. +-commutative 75.6%

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

      11. sub-neg 75.6%

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

      12. metadata-eval 75.6%

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

      13. +-commutative 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around inf 83.2%

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

   if 1.9e92 < t

   1. Initial program 25.5%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in t around inf 98.1%

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

   6. Taylor expanded in t around 0 98.4%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot \left({t}^{2} + {t}^{2}\right) + 2 \cdot {\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{-1 + x}{x + 1}}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x} \cdot \frac{t_m}{l_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.05e+210)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (* (sqrt 2.0) (* (sqrt (* 0.5 x)) (/ t_m 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 tmp;
	if (l_m <= 1.05e+210) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (sqrt((0.5 * x)) * (t_m / l_m));
	}
	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) :: tmp
    if (l_m <= 1.05d+210) then
        tmp = sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (sqrt((0.5d0 * x)) * (t_m / l_m))
    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 tmp;
	if (l_m <= 1.05e+210) {
		tmp = Math.sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((0.5 * x)) * (t_m / l_m));
	}
	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):
	tmp = 0
	if l_m <= 1.05e+210:
		tmp = math.sqrt(((-1.0 + x) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((0.5 * x)) * (t_m / 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)
	tmp = 0.0
	if (l_m <= 1.05e+210)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(0.5 * x)) * Float64(t_m / l_m)));
	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)
	tmp = 0.0;
	if (l_m <= 1.05e+210)
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (sqrt((0.5 * x)) * (t_m / l_m));
	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_] := N[(t$95$s * If[LessEqual[l$95$m, 1.05e+210], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.0499999999999999e210

   1. Initial program 33.8%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in t around inf 47.3%

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

   6. Taylor expanded in t around 0 47.5%

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

   if 1.0499999999999999e210 < l

   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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 52.4%

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

      3. sub-neg 52.4%

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

      4. metadata-eval 52.4%

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

      5. +-commutative 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in x around inf 81.9%

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

      1. *-commutative 81.9%

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

   10. Simplified 81.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x} \cdot \frac{t}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.4 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(0.5, x, -0.5\right)}}{\frac{l_m}{t_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.4e+209)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (/ (sqrt (* 2.0 (fma 0.5 x -0.5))) (/ l_m t_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 tmp;
	if (l_m <= 2.4e+209) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} else {
		tmp = sqrt((2.0 * fma(0.5, x, -0.5))) / (l_m / t_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)
	tmp = 0.0
	if (l_m <= 2.4e+209)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * fma(0.5, x, -0.5))) / Float64(l_m / t_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_] := N[(t$95$s * If[LessEqual[l$95$m, 2.4e+209], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(0.5 * x + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.39999999999999996e209

   1. Initial program 33.8%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in t around inf 47.3%

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

   6. Taylor expanded in t around 0 47.5%

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

   if 2.39999999999999996e209 < l

   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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 52.4%

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

      3. sub-neg 52.4%

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

      4. metadata-eval 52.4%

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

      5. +-commutative 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in x around 0 81.9%

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

      1. associate-*r* 81.8%

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

      2. clear-num 81.8%

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

      3. un-div-inv 81.9%

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

      4. sqrt-unprod 81.9%

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

      5. fma-neg 81.9%

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

      6. metadata-eval 81.9%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(0.5, x, -0.5\right)}}{\frac{\ell}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.2 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.2e+209)
    (sqrt (/ (+ -1.0 x) (+ x 1.0)))
    (* 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 tmp;
	if (l_m <= 1.2e+209) {
		tmp = sqrt(((-1.0 + x) / (x + 1.0)));
	} 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)
	tmp = 0.0
	if (l_m <= 1.2e+209)
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)));
	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_] := N[(t$95$s * If[LessEqual[l$95$m, 1.2e+209], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $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 l < 1.19999999999999998e209

   1. Initial program 33.8%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in t around inf 47.3%

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

   6. Taylor expanded in t around 0 47.5%

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

   if 1.19999999999999998e209 < l

   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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 52.4%

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

      3. sub-neg 52.4%

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

      4. metadata-eval 52.4%

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

      5. +-commutative 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in x around 0 81.9%

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

      1. clear-num 81.9%

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

      2. un-div-inv 81.9%

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

      3. fma-neg 81.9%

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

      4. metadata-eval 81.9%

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

   10. Applied egg-rr 81.9%

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

       1. associate-*r/ 81.9%

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

       2. frac-2neg 81.9%

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

       3. metadata-eval 81.9%

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

       4. fma-neg 81.9%

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

       5. sqrt-unprod 81.9%

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

       6. fma-neg 81.9%

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

       7. metadata-eval 81.9%

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

       8. distribute-neg-frac 81.9%

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

   12. Applied egg-rr 81.9%

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

       1. associate-/r/ 81.9%

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

       2. neg-mul-1 81.9%

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

       3. neg-mul-1 81.9%

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

       4. times-frac 81.9%

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

       5. metadata-eval 81.9%

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

       6. fma-udef 81.9%

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

       7. *-commutative 81.9%

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

       8. fma-def 81.9%

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

   14. Simplified 81.9%

       \[\leadsto \color{blue}{\left(1 \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (sqrt (/ (+ -1.0 x) (+ 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(((-1.0 + x) / (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((((-1.0d0) + x) / (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(((-1.0 + x) / (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(((-1.0 + x) / (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(-1.0 + x) / 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(((-1.0 + x) / (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[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

3. Simplified 31.6%

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

5. Taylor expanded in t around inf 45.0%

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

6. Taylor expanded in t around 0 45.1%

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

7. Final simplification 45.1%

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

Alternative 9: 75.4% 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 1 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 31.7%

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

3. Simplified 31.6%

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

5. Taylor expanded in t around inf 45.0%

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

6. Taylor expanded in x around inf 44.9%

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

7. Final simplification 44.9%

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

Alternative 10: 74.7% 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 1 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 31.7%

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

3. Simplified 31.6%

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

5. Taylor expanded in t around inf 45.0%

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

6. Taylor expanded in x around inf 44.6%

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

7. Final simplification 44.6%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))))