Toniolo and Linder, Equation (7)

Percentage Accurate: 32.2% → 79.3%

Time: 14.1s

Alternatives: 6

Speedup: 85.0×

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: 32.2% 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: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(\left(t\_m \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-137)
    (/ (* (* (* t_m (sqrt 0.5)) (sqrt 2.0)) (sqrt x)) l_m)
    (/ (* t_m (sqrt 2.0)) (* t_m (sqrt (/ (fma 2.0 x 2.0) (+ x -1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-137) {
		tmp = (((t_m * sqrt(0.5)) * sqrt(2.0)) * sqrt(x)) / l_m;
	} else {
		tmp = (t_m * sqrt(2.0)) / (t_m * sqrt((fma(2.0, x, 2.0) / (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)
	tmp = 0.0
	if (t_m <= 1.95e-137)
		tmp = Float64(Float64(Float64(Float64(t_m * sqrt(0.5)) * sqrt(2.0)) * sqrt(x)) / l_m);
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(t_m * sqrt(Float64(fma(2.0, x, 2.0) / Float64(x + -1.0)))));
	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[t$95$m, 1.95e-137], N[(N[(N[(N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e-137

   1. Initial program 4.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 27.2

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 63.4%

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

   if 1.95e-137 < t

   1. Initial program 35.9%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 90.4

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(\left(t\_m \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-137)
    (/ (* (* (* t_m (sqrt 0.5)) (sqrt 2.0)) (sqrt x)) l_m)
    (* (sqrt 2.0) (/ t_m (* t_m (sqrt (/ (fma 2.0 x 2.0) (+ x -1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.95e-137) {
		tmp = (((t_m * sqrt(0.5)) * sqrt(2.0)) * sqrt(x)) / l_m;
	} else {
		tmp = sqrt(2.0) * (t_m / (t_m * sqrt((fma(2.0, x, 2.0) / (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)
	tmp = 0.0
	if (t_m <= 1.95e-137)
		tmp = Float64(Float64(Float64(Float64(t_m * sqrt(0.5)) * sqrt(2.0)) * sqrt(x)) / l_m);
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(t_m * sqrt(Float64(fma(2.0, x, 2.0) / Float64(x + -1.0))))));
	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[t$95$m, 1.95e-137], N[(N[(N[(N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e-137

   1. Initial program 8.4%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 27.2

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 58.6%

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

   if 1.95e-137 < t

   1. Initial program 39.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 85.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left(\left(t \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))))