Toniolo and Linder, Equation (7)

Percentage Accurate: 34.4% → 79.7%

Time: 24.5s

Alternatives: 7

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: 34.4% 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: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t_m}^{2}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;l_m \leq 10^{+86}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t_m}^{2}}{x} + \left(t_2 + \frac{{l_m}^{2}}{x}\right)\right) + \frac{t_2 + {l_m}^{2}}{x}}}\\ \mathbf{elif}\;l_m \leq 2.4 \cdot 10^{+208}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))) (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (*
    t_s
    (if (<= l_m 6.2e+22)
      t_3
      (if (<= l_m 1e+86)
        (*
         t_m
         (/
          (sqrt 2.0)
          (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)))))
        (if (<= l_m 2.4e+208) t_3 (* t_m (/ (sqrt x) 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 = 2.0 * pow(t_m, 2.0);
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (l_m <= 6.2e+22) {
		tmp = t_3;
	} else if (l_m <= 1e+86) {
		tmp = t_m * (sqrt(2.0) / 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 if (l_m <= 2.4e+208) {
		tmp = t_3;
	} else {
		tmp = t_m * (sqrt(x) / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (l_m <= 6.2d+22) then
        tmp = t_3
    else if (l_m <= 1d+86) then
        tmp = t_m * (sqrt(2.0d0) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else if (l_m <= 2.4d+208) then
        tmp = t_3
    else
        tmp = t_m * (sqrt(x) / 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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (l_m <= 6.2e+22) {
		tmp = t_3;
	} else if (l_m <= 1e+86) {
		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))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else if (l_m <= 2.4e+208) {
		tmp = t_3;
	} else {
		tmp = t_m * (Math.sqrt(x) / 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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if l_m <= 6.2e+22:
		tmp = t_3
	elif l_m <= 1e+86:
		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))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	elif l_m <= 2.4e+208:
		tmp = t_3
	else:
		tmp = t_m * (math.sqrt(x) / 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(2.0 * (t_m ^ 2.0))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (l_m <= 6.2e+22)
		tmp = t_3;
	elseif (l_m <= 1e+86)
		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(t_2 + (l_m ^ 2.0)) / x)))));
	elseif (l_m <= 2.4e+208)
		tmp = t_3;
	else
		tmp = Float64(t_m * Float64(sqrt(x) / 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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (l_m <= 6.2e+22)
		tmp = t_3;
	elseif (l_m <= 1e+86)
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	elseif (l_m <= 2.4e+208)
		tmp = t_3;
	else
		tmp = t_m * (sqrt(x) / 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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 6.2e+22], t$95$3, If[LessEqual[l$95$m, 1e+86], 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[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.4e+208], t$95$3, N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.2000000000000004e22 or 1e86 < l < 2.39999999999999987e208

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around 0 36.4%

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

      1. +-commutative 36.4%

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

      2. sub-neg 36.4%

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

      3. metadata-eval 36.4%

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

      4. +-commutative 36.4%

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

   7. Simplified 36.4%

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

      1. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      2. sqrt-undiv 36.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{2}}}}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      3. metadata-eval 36.3%

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

      4. sqrt-div 36.4%

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

      5. clear-num 36.4%

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

      6. +-commutative 36.4%

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

   9. Applied egg-rr 36.4%

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

   if 6.2000000000000004e22 < l < 1e86

   1. Initial program 23.9%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in x around inf 66.6%

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

   if 2.39999999999999987e208 < 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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 29.7%

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

      3. sub-neg 29.7%

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

      4. metadata-eval 29.7%

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

      5. +-commutative 29.7%

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

      6. sub-neg 29.7%

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

      7. metadata-eval 29.7%

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

      8. +-commutative 29.7%

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

   7. Simplified 29.7%

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

      1. expm1-log1p-u 29.7%

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

      2. expm1-udef 22.5%

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

   9. Applied egg-rr 22.5%

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

       1. expm1-def 29.7%

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

       2. expm1-log1p 29.7%

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

       3. associate-/r/ 32.7%

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

       4. associate-+l+ 2.3%

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

       5. +-commutative 2.3%

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

       6. +-commutative 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in x around inf 87.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;\ell \leq 10^{+86}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x + -1}{x + 1}\\ t_3 := \sqrt{t_2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.48 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;l_m \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t_m}{\sqrt{\mathsf{fma}\left(2, \frac{{t_m}^{2}}{t_2}, 2 \cdot \frac{{l_m}^{2}}{x}\right)}}\\ \mathbf{elif}\;l_m \leq 1.45 \cdot 10^{+207}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (+ x -1.0) (+ x 1.0))) (t_3 (sqrt t_2)))
   (*
    t_s
    (if (<= l_m 1.48e+22)
      t_3
      (if (<= l_m 1.06e+86)
        (*
         (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 1.45e+207) t_3 (* t_m (/ (sqrt x) 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 t_3 = sqrt(t_2);
	double tmp;
	if (l_m <= 1.48e+22) {
		tmp = t_3;
	} else if (l_m <= 1.06e+86) {
		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 <= 1.45e+207) {
		tmp = t_3;
	} else {
		tmp = t_m * (sqrt(x) / 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))
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (l_m <= 1.48e+22)
		tmp = t_3;
	elseif (l_m <= 1.06e+86)
		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 <= 1.45e+207)
		tmp = t_3;
	else
		tmp = Float64(t_m * Float64(sqrt(x) / 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]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.48e+22], t$95$3, If[LessEqual[l$95$m, 1.06e+86], 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, 1.45e+207], t$95$3, N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.48e22 or 1.06e86 < l < 1.44999999999999999e207

   1. Initial program 34.3%

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

   3. Simplified 34.3%

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

   5. Taylor expanded in l around 0 36.4%

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

      1. +-commutative 36.4%

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

      2. sub-neg 36.4%

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

      3. metadata-eval 36.4%

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

      4. +-commutative 36.4%

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

   7. Simplified 36.4%

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

      1. associate-/r* 36.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      2. sqrt-undiv 36.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{2}}}}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      3. metadata-eval 36.3%

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

      4. sqrt-div 36.4%

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

      5. clear-num 36.4%

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

      6. +-commutative 36.4%

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

   9. Applied egg-rr 36.4%

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

   if 1.48e22 < l < 1.06e86

   1. Initial program 23.9%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in l around 0 3.2%

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

      1. fma-def 3.2%

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

      2. associate-/l* 24.4%

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

      3. +-commutative 24.4%

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

      4. sub-neg 24.4%

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

      5. metadata-eval 24.4%

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

      6. +-commutative 24.4%

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

      7. associate--l+ 31.9%

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

      8. sub-neg 31.9%

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

      9. metadata-eval 31.9%

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

      10. +-commutative 31.9%

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

      11. sub-neg 31.9%

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

      12. metadata-eval 31.9%

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

      13. +-commutative 31.9%

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

   7. Simplified 31.9%

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

   8. Taylor expanded in x around inf 66.7%

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

   if 1.44999999999999999e207 < 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%

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

   5. Taylor expanded in l around inf 1.5%

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

      1. *-commutative 1.5%

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

      2. associate--l+ 29.7%

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

      3. sub-neg 29.7%

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

      4. metadata-eval 29.7%

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

      5. +-commutative 29.7%

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

      6. sub-neg 29.7%

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

      7. metadata-eval 29.7%

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

      8. +-commutative 29.7%

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

   7. Simplified 29.7%

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

      1. expm1-log1p-u 29.7%

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

      2. expm1-udef 22.5%

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

   9. Applied egg-rr 22.5%

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

       1. expm1-def 29.7%

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

       2. expm1-log1p 29.7%

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

       3. associate-/r/ 32.7%

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

       4. associate-+l+ 2.3%

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

       5. +-commutative 2.3%

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

       6. +-commutative 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in x around inf 87.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.48 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{\frac{x + -1}{x + 1}}, 2 \cdot \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.22 \cdot 10^{+64} \lor \neg \left(l_m \leq 10^{+86}\right) \land l_m \leq 1.9 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \]

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 (or (<= l_m 1.22e+64) (and (not (<= l_m 1e+86)) (<= l_m 1.9e+208)))
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt x) 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 <= 1.22e+64) || (!(l_m <= 1e+86) && (l_m <= 1.9e+208))) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(x) / 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 <= 1.22d+64) .or. (.not. (l_m <= 1d+86)) .and. (l_m <= 1.9d+208)) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(x) / 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 <= 1.22e+64) || (!(l_m <= 1e+86) && (l_m <= 1.9e+208))) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / 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 <= 1.22e+64) or (not (l_m <= 1e+86) and (l_m <= 1.9e+208)):
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / 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 <= 1.22e+64) || (!(l_m <= 1e+86) && (l_m <= 1.9e+208)))
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / 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 <= 1.22e+64) || (~((l_m <= 1e+86)) && (l_m <= 1.9e+208)))
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(x) / 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[Or[LessEqual[l$95$m, 1.22e+64], And[N[Not[LessEqual[l$95$m, 1e+86]], $MachinePrecision], LessEqual[l$95$m, 1.9e+208]]], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.21999999999999994e64 or 1e86 < l < 1.9000000000000001e208

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in l around 0 35.8%

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

      1. +-commutative 35.8%

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

      2. sub-neg 35.8%

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

      3. metadata-eval 35.8%

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

      4. +-commutative 35.8%

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

   7. Simplified 35.8%

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

      1. associate-/r* 35.8%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      2. sqrt-undiv 35.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{2}}}}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      3. metadata-eval 35.8%

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

      4. sqrt-div 35.8%

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

      5. clear-num 35.8%

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

      6. +-commutative 35.8%

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

   9. Applied egg-rr 35.8%

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

   if 1.21999999999999994e64 < l < 1e86 or 1.9000000000000001e208 < l

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 27.7%

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

      3. sub-neg 27.7%

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

      4. metadata-eval 27.7%

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

      5. +-commutative 27.7%

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

      6. sub-neg 27.7%

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

      7. metadata-eval 27.7%

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

      8. +-commutative 27.7%

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

   7. Simplified 27.7%

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

      1. expm1-log1p-u 27.7%

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

      2. expm1-udef 20.4%

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

   9. Applied egg-rr 20.4%

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

       1. expm1-def 27.7%

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

       2. expm1-log1p 27.7%

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

       3. associate-/r/ 30.4%

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

       4. associate-+l+ 2.3%

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

       5. +-commutative 2.3%

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

       6. +-commutative 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in x around inf 83.3%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.22 \cdot 10^{+64} \lor \neg \left(\ell \leq 10^{+86}\right) \land \ell \leq 1.9 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2 \cdot 10^{+63} \lor \neg \left(l_m \leq 10^{+86}\right) \land l_m \leq 2 \cdot 10^{+209}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x}}{l_m}\\ \end{array} \end{array} \]

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 (or (<= l_m 2e+63) (and (not (<= l_m 1e+86)) (<= l_m 2e+209)))
    (+ 1.0 (/ -1.0 x))
    (* t_m (/ (sqrt x) 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 <= 2e+63) || (!(l_m <= 1e+86) && (l_m <= 2e+209))) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (sqrt(x) / 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 <= 2d+63) .or. (.not. (l_m <= 1d+86)) .and. (l_m <= 2d+209)) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * (sqrt(x) / 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 <= 2e+63) || (!(l_m <= 1e+86) && (l_m <= 2e+209))) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / 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 <= 2e+63) or (not (l_m <= 1e+86) and (l_m <= 2e+209)):
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * (math.sqrt(x) / 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 <= 2e+63) || (!(l_m <= 1e+86) && (l_m <= 2e+209)))
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / 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 <= 2e+63) || (~((l_m <= 1e+86)) && (l_m <= 2e+209)))
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * (sqrt(x) / 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[Or[LessEqual[l$95$m, 2e+63], And[N[Not[LessEqual[l$95$m, 1e+86]], $MachinePrecision], LessEqual[l$95$m, 2e+209]]], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.00000000000000012e63 or 1e86 < l < 2.0000000000000001e209

   1. Initial program 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in l around 0 35.8%

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

      1. +-commutative 35.8%

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

      2. sub-neg 35.8%

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

      3. metadata-eval 35.8%

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

      4. +-commutative 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in x around inf 35.7%

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

   if 2.00000000000000012e63 < l < 1e86 or 2.0000000000000001e209 < l

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in l around inf 1.7%

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

      1. *-commutative 1.7%

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

      2. associate--l+ 27.7%

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

      3. sub-neg 27.7%

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

      4. metadata-eval 27.7%

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

      5. +-commutative 27.7%

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

      6. sub-neg 27.7%

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

      7. metadata-eval 27.7%

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

      8. +-commutative 27.7%

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

   7. Simplified 27.7%

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

      1. expm1-log1p-u 27.7%

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

      2. expm1-udef 20.4%

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

   9. Applied egg-rr 20.4%

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

       1. expm1-def 27.7%

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

       2. expm1-log1p 27.7%

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

       3. associate-/r/ 30.4%

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

       4. associate-+l+ 2.3%

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

       5. +-commutative 2.3%

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

       6. +-commutative 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in x around inf 83.3%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+63} \lor \neg \left(\ell \leq 10^{+86}\right) \land \ell \leq 2 \cdot 10^{+209}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;l_m \cdot l_m \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t_m \cdot \frac{\sqrt{x + -1}}{l_m}\\ \end{array} \end{array} \]

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 l_m) 2e+114)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt (+ x -1.0)) 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 * l_m) <= 2e+114) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt((x + -1.0)) / 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 * l_m) <= 2d+114) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt((x + (-1.0d0))) / 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 * l_m) <= 2e+114) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt((x + -1.0)) / 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 * l_m) <= 2e+114:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt((x + -1.0)) / 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 (Float64(l_m * l_m) <= 2e+114)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(Float64(x + -1.0)) / 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 * l_m) <= 2e+114)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt((x + -1.0)) / 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[N[(l$95$m * l$95$m), $MachinePrecision], 2e+114], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2e114

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in l around 0 39.1%

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

      1. +-commutative 39.1%

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

      2. sub-neg 39.1%

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

      3. metadata-eval 39.1%

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

      4. +-commutative 39.1%

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

   7. Simplified 39.1%

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

      1. associate-/r* 39.1%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      2. sqrt-undiv 39.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{2}}}}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      3. metadata-eval 39.1%

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

      4. sqrt-div 39.1%

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

      5. clear-num 39.1%

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

      6. +-commutative 39.1%

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

   9. Applied egg-rr 39.1%

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

   if 2e114 < (*.f64 l l)

   1. Initial program 0.6%

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

   3. Simplified 0.6%

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

   5. Taylor expanded in l around inf 3.6%

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

      1. *-commutative 3.6%

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

      2. associate--l+ 26.9%

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

      3. sub-neg 26.9%

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

      4. metadata-eval 26.9%

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

      5. +-commutative 26.9%

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

      6. sub-neg 26.9%

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

      7. metadata-eval 26.9%

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

      8. +-commutative 26.9%

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

   7. Simplified 26.9%

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

      1. expm1-log1p-u 26.6%

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

      2. expm1-udef 23.6%

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

   9. Applied egg-rr 23.5%

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

       1. expm1-def 26.4%

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

       2. expm1-log1p 26.8%

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

       3. associate-/r/ 27.4%

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

       4. associate-+l+ 4.2%

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

       5. +-commutative 4.2%

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

       6. +-commutative 4.2%

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

   11. Simplified 4.2%

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

   12. Taylor expanded in x around 0 43.7%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x + -1}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% 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 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in l around 0 34.3%

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

   1. +-commutative 34.3%

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

   2. sub-neg 34.3%

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

   3. metadata-eval 34.3%

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

   4. +-commutative 34.3%

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

7. Simplified 34.3%

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

8. Taylor expanded in x around inf 34.2%

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

9. Final simplification 34.2%

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

Alternative 7: 76.1% 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 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in l around 0 34.3%

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

   1. +-commutative 34.3%

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

   2. sub-neg 34.3%

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

   3. metadata-eval 34.3%

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

   4. +-commutative 34.3%

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

7. Simplified 34.3%

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

8. Taylor expanded in x around inf 33.8%

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

9. Final simplification 33.8%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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)))))