Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 80.2%

Time: 13.0s

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: 33.2% 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: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.42 \cdot 10^{+221}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{l\_m}{\left(x \cdot t\_m\right) \cdot \sqrt{2}} \cdot \left(2 \cdot l\_m\right), 0.5, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0.5, x, -0.5\right) \cdot 2} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= l_m 1.42e+221)
      (/ t_2 (fma (* (/ l_m (* (* x t_m) (sqrt 2.0))) (* 2.0 l_m)) 0.5 t_2))
      (/ (* (sqrt (* (fma 0.5 x -0.5) 2.0)) t_m) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (l_m <= 1.42e+221) {
		tmp = t_2 / fma(((l_m / ((x * t_m) * sqrt(2.0))) * (2.0 * l_m)), 0.5, t_2);
	} else {
		tmp = (sqrt((fma(0.5, x, -0.5) * 2.0)) * t_m) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (l_m <= 1.42e+221)
		tmp = Float64(t_2 / fma(Float64(Float64(l_m / Float64(Float64(x * t_m) * sqrt(2.0))) * Float64(2.0 * l_m)), 0.5, t_2));
	else
		tmp = Float64(Float64(sqrt(Float64(fma(0.5, x, -0.5) * 2.0)) * t_m) / 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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.42e+221], N[(t$95$2 / N[(N[(N[(l$95$m / N[(N[(x * t$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(0.5 * x + -0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.42000000000000001e221

   1. Initial program 39.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in l around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2 \cdot \frac{{\ell}^{2}}{t \cdot \left(x \cdot \sqrt{2}\right)}, \frac{1}{2}, \sqrt{2} \cdot t\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.8%

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

   10. Applied rewrites 86.2%

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

   if 1.42000000000000001e221 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 79.1%

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

   13. Applied rewrites 89.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{+221}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(\frac{\ell}{\left(x \cdot t\right) \cdot \sqrt{2}} \cdot \left(2 \cdot \ell\right), 0.5, t \cdot \sqrt{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0.5, x, -0.5\right) \cdot 2} \cdot t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.45 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(0.5, x, -0.5\right) \cdot 2} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

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 (<= l_m 2.45e+215)
    (sqrt (/ (- x 1.0) (- x -1.0)))
    (/ (* (sqrt (* (fma 0.5 x -0.5) 2.0)) t_m) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.45e+215) {
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	} else {
		tmp = (sqrt((fma(0.5, x, -0.5) * 2.0)) * t_m) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.45e+215)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(fma(0.5, x, -0.5) * 2.0)) * t_m) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.45e+215], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(N[(0.5 * x + -0.5), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.4500000000000001e215

   1. Initial program 37.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 81.4%

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

   if 2.4500000000000001e215 < l

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.4%

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

   13. Applied rewrites 64.7%

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

Reproduce

herbie shell --seed 2024229 
(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)))))