Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 83.8%

Time: 14.0s

Alternatives: 9

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: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \frac{\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 6.8e-194)
    (/
     (* t_m (sqrt 2.0))
     (*
      l_m
      (fma
       (sqrt 2.0)
       (sqrt (/ 1.0 x))
       (/ (sqrt (/ 1.0 (* x (* x x)))) (sqrt 2.0)))))
    (if (<= t_m 1.05e-171)
      1.0
      (if (<= t_m 5.2e+36)
        (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
        (+ 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) {
	double tmp;
	if (t_m <= 6.8e-194) {
		tmp = (t_m * sqrt(2.0)) / (l_m * fma(sqrt(2.0), sqrt((1.0 / x)), (sqrt((1.0 / (x * (x * x)))) / sqrt(2.0))));
	} else if (t_m <= 1.05e-171) {
		tmp = 1.0;
	} else if (t_m <= 5.2e+36) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 6.8e-194)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * fma(sqrt(2.0), sqrt(Float64(1.0 / x)), Float64(sqrt(Float64(1.0 / Float64(x * Float64(x * x)))) / sqrt(2.0)))));
	elseif (t_m <= 1.05e-171)
		tmp = 1.0;
	elseif (t_m <= 5.2e+36)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-194], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e-171], 1.0, If[LessEqual[t$95$m, 5.2e+36], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.80000000000000018e-194

   1. Initial program 24.3%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}} + \frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{1}{{x}^{3}}}}} \]
   7. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)} + \frac{\ell}{\sqrt{2}} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\frac{\ell \cdot \sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right) + \color{blue}{\ell \cdot \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}} + \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}} + \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \color{blue}{\mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\color{blue}{\sqrt{2}}, \sqrt{\frac{1}{x}}, \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \color{blue}{\frac{\sqrt{\frac{1}{{x}^{3}}}}{\sqrt{2}}}\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \frac{\color{blue}{\sqrt{\frac{1}{{x}^{3}}}}}{\sqrt{2}}\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \frac{\sqrt{\color{blue}{\frac{1}{{x}^{3}}}}}{\sqrt{2}}\right)} \]

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-sqrt.f64 21.6

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

   8. Applied rewrites 21.6%

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

   if 6.80000000000000018e-194 < t < 1.05e-171

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 98.4

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.05e-171 < t < 5.2000000000000003e36

   1. Initial program 53.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. div-inv N/A

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

      15. associate-*l* N/A

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

   7. Applied rewrites 92.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

   9. Applied rewrites 92.8%

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

   if 5.2000000000000003e36 < t

   1. Initial program 36.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 95.8

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

   8. Applied rewrites 95.8%

      \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \mathsf{fma}\left(\sqrt{2}, \sqrt{\frac{1}{x}}, \frac{\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}}{\sqrt{2}}\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 6.8e-194)
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
    (if (<= t_m 1.05e-171)
      1.0
      (if (<= t_m 5.2e+36)
        (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
        (+ 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) {
	double tmp;
	if (t_m <= 6.8e-194) {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((2.0 + (2.0 / x)) / x)));
	} else if (t_m <= 1.05e-171) {
		tmp = 1.0;
	} else if (t_m <= 5.2e+36) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 6.8e-194)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	elseif (t_m <= 1.05e-171)
		tmp = 1.0;
	elseif (t_m <= 5.2e+36)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-194], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e-171], 1.0, If[LessEqual[t$95$m, 5.2e+36], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.80000000000000018e-194

   1. Initial program 24.3%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-+l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 21.6

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

   8. Applied rewrites 21.6%

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

   if 6.80000000000000018e-194 < t < 1.05e-171

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 98.4

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.05e-171 < t < 5.2000000000000003e36

   1. Initial program 53.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. div-inv N/A

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

      15. associate-*l* N/A

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

   7. Applied rewrites 92.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

   9. Applied rewrites 92.8%

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

   if 5.2000000000000003e36 < t

   1. Initial program 36.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 95.8

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

   8. Applied rewrites 95.8%

      \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-194}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_m}{\sqrt{\mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 1.4e-194)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 1.05e-171)
      1.0
      (if (<= t_m 5.2e+36)
        (/ t_m (sqrt (fma t_m t_m (/ (fma 2.0 (* t_m t_m) (* l_m l_m)) x))))
        (+ 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) {
	double tmp;
	if (t_m <= 1.4e-194) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.05e-171) {
		tmp = 1.0;
	} else if (t_m <= 5.2e+36) {
		tmp = t_m / sqrt(fma(t_m, t_m, (fma(2.0, (t_m * t_m), (l_m * l_m)) / x)));
	} else {
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 1.4e-194)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.05e-171)
		tmp = 1.0;
	elseif (t_m <= 5.2e+36)
		tmp = Float64(t_m / sqrt(fma(t_m, t_m, Float64(fma(2.0, Float64(t_m * t_m), Float64(l_m * l_m)) / x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.4e-194], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.05e-171], 1.0, If[LessEqual[t$95$m, 5.2e+36], N[(t$95$m / N[Sqrt[N[(t$95$m * t$95$m + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.40000000000000006e-194

   1. Initial program 24.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 24.3

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

   8. Applied rewrites 24.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 23.7

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

   10. Applied rewrites 23.7%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 21.5

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

   13. Applied rewrites 21.5%

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

   if 1.40000000000000006e-194 < t < 1.05e-171

   1. Initial program 3.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 98.4

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.05e-171 < t < 5.2000000000000003e36

   1. Initial program 53.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. div-inv N/A

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

      15. associate-*l* N/A

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-frac-neg N/A

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

   7. Applied rewrites 92.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

   9. Applied rewrites 92.8%

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

   if 5.2000000000000003e36 < t

   1. Initial program 36.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 95.8

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

   8. Applied rewrites 95.8%

      \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

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 8.5e-138)
    (/ (* t_m (sqrt x)) l_m)
    (sqrt (/ (+ x -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) {
	double tmp;
	if (t_m <= 8.5e-138) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + 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) :: tmp
    if (t_m <= 8.5d-138) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + 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 tmp;
	if (t_m <= 8.5e-138) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + 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):
	tmp = 0
	if t_m <= 8.5e-138:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + 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)
	tmp = 0.0
	if (t_m <= 8.5e-138)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + 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)
	tmp = 0.0;
	if (t_m <= 8.5e-138)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = sqrt(((x + -1.0) / (1.0 + 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_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-138], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000035e-138

   1. Initial program 23.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 26.2

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

   10. Applied rewrites 26.2%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 23.4

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

   13. Applied rewrites 23.4%

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

   if 8.50000000000000035e-138 < t

   1. Initial program 45.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 8.5e-138) (/ (* t_m (sqrt x)) l_m) (+ 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) {
	double tmp;
	if (t_m <= 8.5e-138) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + (-1.0 / 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) :: tmp
    if (t_m <= 8.5d-138) then
        tmp = (t_m * sqrt(x)) / l_m
    else
        tmp = 1.0d0 + ((-1.0d0) / 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 tmp;
	if (t_m <= 8.5e-138) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else {
		tmp = 1.0 + (-1.0 / 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):
	tmp = 0
	if t_m <= 8.5e-138:
		tmp = (t_m * math.sqrt(x)) / l_m
	else:
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 8.5e-138)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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)
	tmp = 0.0;
	if (t_m <= 8.5e-138)
		tmp = (t_m * sqrt(x)) / l_m;
	else
		tmp = 1.0 + (-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-138], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000035e-138

   1. Initial program 23.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 26.2

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

   10. Applied rewrites 26.2%

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

   11. Taylor expanded in t around 0

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

       1. associate-*l/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 23.4

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

   13. Applied rewrites 23.4%

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

   if 8.50000000000000035e-138 < t

   1. Initial program 45.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 88.4

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

   8. Applied rewrites 88.4%

      \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 8.5e-138) (* t_m (/ (sqrt x) l_m)) (+ 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) {
	double tmp;
	if (t_m <= 8.5e-138) {
		tmp = t_m * (sqrt(x) / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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) :: tmp
    if (t_m <= 8.5d-138) then
        tmp = t_m * (sqrt(x) / l_m)
    else
        tmp = 1.0d0 + ((-1.0d0) / 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 tmp;
	if (t_m <= 8.5e-138) {
		tmp = t_m * (Math.sqrt(x) / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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):
	tmp = 0
	if t_m <= 8.5e-138:
		tmp = t_m * (math.sqrt(x) / l_m)
	else:
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 8.5e-138)
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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)
	tmp = 0.0;
	if (t_m <= 8.5e-138)
		tmp = t_m * (sqrt(x) / l_m);
	else
		tmp = 1.0 + (-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-138], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000035e-138

   1. Initial program 23.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 26.2

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

   10. Applied rewrites 26.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 26.2

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

   12. Applied rewrites 23.3%

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

   if 8.50000000000000035e-138 < t

   1. Initial program 45.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 88.4

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

   8. Applied rewrites 88.4%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 8.5e-138) (* (sqrt x) (/ t_m l_m)) (+ 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) {
	double tmp;
	if (t_m <= 8.5e-138) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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) :: tmp
    if (t_m <= 8.5d-138) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = 1.0d0 + ((-1.0d0) / 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 tmp;
	if (t_m <= 8.5e-138) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + (-1.0 / 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):
	tmp = 0
	if t_m <= 8.5e-138:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = 1.0 + (-1.0 / 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)
	tmp = 0.0
	if (t_m <= 8.5e-138)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = Float64(1.0 + Float64(-1.0 / 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)
	tmp = 0.0;
	if (t_m <= 8.5e-138)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = 1.0 + (-1.0 / 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_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-138], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000035e-138

   1. Initial program 23.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 26.2

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

   10. Applied rewrites 26.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. sqrt-div N/A

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

       5. associate-*r/ N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. sqrt-prod N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. sqrt-pow1 N/A

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

       12. metadata-eval N/A

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

       13. unpow1 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. times-frac N/A

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

       16. lift-sqrt.f64 N/A

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

       17. sqrt-div N/A

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. associate-/l* N/A

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

       21. metadata-eval N/A

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

   12. Applied rewrites 20.2%

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

   if 8.50000000000000035e-138 < t

   1. Initial program 45.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
   7. Step-by-step derivation

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 88.4

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

   8. Applied rewrites 88.4%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 5.7× 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 #s(literal 1 binary64) 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 32.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 39.1

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

5. Applied rewrites 39.1%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
7. Step-by-step derivation

   1. sub-neg N/A

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

   2. lower-+.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 38.7

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

8. Applied rewrites 38.7%

   \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
9. Add Preprocessing

Alternative 9: 74.6% accurate, 85.0× 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 #s(literal 1 binary64) 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 32.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 37.4

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{0.5}} \]

5. Applied rewrites 37.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval 38.0

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

7. Applied rewrites 38.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))