Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 79.2%

Time: 14.5s

Alternatives: 8

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: 33.1% 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: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{+276}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= (* l_m l_m) 1e+276)
      (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))
      (/ t_2 (* l_m (sqrt (/ 2.0 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) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if ((l_m * l_m) <= 1e+276) {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = t_2 / (l_m * sqrt((2.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if ((l_m * l_m) <= 1d+276) then
        tmp = t_2 / (t_2 * sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    else
        tmp = t_2 / (l_m * sqrt((2.0d0 / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if ((l_m * l_m) <= 1e+276) {
		tmp = t_2 / (t_2 * Math.sqrt(((x + 1.0) / (x + -1.0))));
	} else {
		tmp = t_2 / (l_m * Math.sqrt((2.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if (l_m * l_m) <= 1e+276:
		tmp = t_2 / (t_2 * math.sqrt(((x + 1.0) / (x + -1.0))))
	else:
		tmp = t_2 / (l_m * math.sqrt((2.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+276)
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	else
		tmp = Float64(t_2 / Float64(l_m * sqrt(Float64(2.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if ((l_m * l_m) <= 1e+276)
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	else
		tmp = t_2 / (l_m * sqrt((2.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+276], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e276

   1. Initial program 40.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 83.2

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

   5. Applied rewrites 83.2%

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

   if 1.0000000000000001e276 < (*.f64 l 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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. sub-neg N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{+276}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{l\_m \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= (* l_m l_m) 1e+276)
      (/ t_2 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0)))))
      (/ t_2 (* l_m (sqrt (/ 2.0 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) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if ((l_m * l_m) <= 1e+276) {
		tmp = t_2 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	} else {
		tmp = t_2 / (l_m * sqrt((2.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 1e+276)
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	else
		tmp = Float64(t_2 / Float64(l_m * sqrt(Float64(2.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 1e+276], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(l$95$m * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e276

   1. Initial program 42.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 85.4

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

   5. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 85.3%

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

   9. Applied rewrites 85.3%

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

   if 1.0000000000000001e276 < (*.f64 l l)

   1. Initial program 0.6%

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. sub-neg N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 3.0

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

   7. Applied rewrites 3.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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)))))