Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 82.3%

Time: 14.9s

Alternatives: 12

Speedup: 85.0×

Specification

Math

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

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 32.9% 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: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\mathsf{fma}\left(\ell \cdot \ell, -2, \frac{\left(\left(\ell \cdot \ell\right) \cdot -2 - t\_2\right) - t\_2}{x}\right)}{-x} + \frac{\left(2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(x + 1\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (* l l) x)) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.3e-230)
      (/ t_3 (fma 0.5 (/ (* 2.0 (* l l)) (* t_m (* (sqrt 2.0) x))) t_3))
      (if (<= t_m 4e-7)
        (/
         t_3
         (sqrt
          (+
           (/ (fma (* l l) -2.0 (/ (- (- (* (* l l) -2.0) t_2) t_2) x)) (- x))
           (/ (* (* 2.0 (* t_m t_m)) (+ x 1.0)) (+ x -1.0)))))
        (/ t_3 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = (l * l) / x;
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.3e-230) {
		tmp = t_3 / fma(0.5, ((2.0 * (l * l)) / (t_m * (sqrt(2.0) * x))), t_3);
	} else if (t_m <= 4e-7) {
		tmp = t_3 / sqrt(((fma((l * l), -2.0, (((((l * l) * -2.0) - t_2) - t_2) / x)) / -x) + (((2.0 * (t_m * t_m)) * (x + 1.0)) / (x + -1.0))));
	} else {
		tmp = t_3 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.3e-230)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(sqrt(2.0) * x))), t_3));
	elseif (t_m <= 4e-7)
		tmp = Float64(t_3 / sqrt(Float64(Float64(fma(Float64(l * l), -2.0, Float64(Float64(Float64(Float64(Float64(l * l) * -2.0) - t_2) - t_2) / x)) / Float64(-x)) + Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) * Float64(x + 1.0)) / Float64(x + -1.0)))));
	else
		tmp = Float64(t_3 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-230], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e-7], N[(t$95$3 / N[Sqrt[N[(N[(N[(N[(l * l), $MachinePrecision] * -2.0 + N[(N[(N[(N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$2), $MachinePrecision] - t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] + N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-230

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 11.3%

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

   if 1.3000000000000001e-230 < t < 3.9999999999999998e-7

   1. Initial program 37.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 82.6%

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

   if 3.9999999999999998e-7 < t

   1. Initial program 36.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 92.0

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

   9. Applied rewrites 92.0%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(\ell \cdot \ell, -2, \frac{\left(\left(\ell \cdot \ell\right) \cdot -2 - \frac{\ell \cdot \ell}{x}\right) - \frac{\ell \cdot \ell}{x}}{x}\right)}{-x} + \frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(x + 1\right)}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{\left(2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(x + 1\right)}{x + -1} + \frac{t\_2 + \left(t\_2 - \left(\ell \cdot \ell\right) \cdot -2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (/ (* l l) x)) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.3e-230)
      (/ t_3 (fma 0.5 (/ (* 2.0 (* l l)) (* t_m (* (sqrt 2.0) x))) t_3))
      (if (<= t_m 4e-7)
        (/
         t_3
         (sqrt
          (+
           (/ (* (* 2.0 (* t_m t_m)) (+ x 1.0)) (+ x -1.0))
           (/ (+ t_2 (- t_2 (* (* l l) -2.0))) x))))
        (/ t_3 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = (l * l) / x;
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.3e-230) {
		tmp = t_3 / fma(0.5, ((2.0 * (l * l)) / (t_m * (sqrt(2.0) * x))), t_3);
	} else if (t_m <= 4e-7) {
		tmp = t_3 / sqrt(((((2.0 * (t_m * t_m)) * (x + 1.0)) / (x + -1.0)) + ((t_2 + (t_2 - ((l * l) * -2.0))) / x)));
	} else {
		tmp = t_3 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.3e-230)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(sqrt(2.0) * x))), t_3));
	elseif (t_m <= 4e-7)
		tmp = Float64(t_3 / sqrt(Float64(Float64(Float64(Float64(2.0 * Float64(t_m * t_m)) * Float64(x + 1.0)) / Float64(x + -1.0)) + Float64(Float64(t_2 + Float64(t_2 - Float64(Float64(l * l) * -2.0))) / x))));
	else
		tmp = Float64(t_3 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-230], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e-7], N[(t$95$3 / N[Sqrt[N[(N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[(t$95$2 - N[(N[(l * l), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-230

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 11.3%

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

   if 1.3000000000000001e-230 < t < 3.9999999999999998e-7

   1. Initial program 37.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 82.2%

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

   if 3.9999999999999998e-7 < t

   1. Initial program 36.0%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 92.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 92.0

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

   9. Applied rewrites 92.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(x + 1\right)}{x + -1} + \frac{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} - \left(\ell \cdot \ell\right) \cdot -2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.3e-230)
      (/ t_2 (fma 0.5 (/ (* 2.0 (* l l)) (* t_m (* (sqrt 2.0) x))) t_2))
      (if (<= t_m 5.1e+14)
        (/
         t_2
         (sqrt
          (+
           (fma 2.0 (+ (* t_m t_m) (/ (* t_m t_m) x)) (/ (* l l) x))
           (/ (fma 2.0 (* t_m t_m) (* l l)) x))))
        (/ t_2 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.3e-230) {
		tmp = t_2 / fma(0.5, ((2.0 * (l * l)) / (t_m * (sqrt(2.0) * x))), t_2);
	} else if (t_m <= 5.1e+14) {
		tmp = t_2 / sqrt((fma(2.0, ((t_m * t_m) + ((t_m * t_m) / x)), ((l * l) / x)) + (fma(2.0, (t_m * t_m), (l * l)) / x)));
	} else {
		tmp = t_2 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.3e-230)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(sqrt(2.0) * x))), t_2));
	elseif (t_m <= 5.1e+14)
		tmp = Float64(t_2 / sqrt(Float64(fma(2.0, Float64(Float64(t_m * t_m) + Float64(Float64(t_m * t_m) / x)), Float64(Float64(l * l) / x)) + Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x))));
	else
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-230], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+14], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-230

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 11.3%

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

   if 1.3000000000000001e-230 < t < 5.1e14

   1. Initial program 36.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Applied rewrites 81.5%

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

   if 5.1e14 < t

   1. Initial program 36.5%

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

   3. Taylor expanded in 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 91.6

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

   5. Applied rewrites 91.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 91.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 91.6

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

   9. Applied rewrites 91.6%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t + \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\sqrt{2} \cdot x\right)}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{t\_m \cdot t\_m}{x}\right), \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 1.3e-230)
      (/ t_2 (fma 0.5 (/ (* 2.0 (* l l)) (* t_m (* (sqrt 2.0) x))) t_2))
      (if (<= t_m 5.1e+14)
        (/
         t_2
         (sqrt
          (+
           (/ (fma 2.0 (* t_m t_m) (* l l)) x)
           (fma 2.0 (fma t_m t_m (/ (* t_m t_m) x)) (/ (* l l) x)))))
        (/ t_2 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.3e-230) {
		tmp = t_2 / fma(0.5, ((2.0 * (l * l)) / (t_m * (sqrt(2.0) * x))), t_2);
	} else if (t_m <= 5.1e+14) {
		tmp = t_2 / sqrt(((fma(2.0, (t_m * t_m), (l * l)) / x) + fma(2.0, fma(t_m, t_m, ((t_m * t_m) / x)), ((l * l) / x))));
	} else {
		tmp = t_2 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 1.3e-230)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(sqrt(2.0) * x))), t_2));
	elseif (t_m <= 5.1e+14)
		tmp = Float64(t_2 / sqrt(Float64(Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x) + fma(2.0, fma(t_m, t_m, Float64(Float64(t_m * t_m) / x)), Float64(Float64(l * l) / x)))));
	else
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-230], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+14], N[(t$95$2 / N[Sqrt[N[(N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3000000000000001e-230

   1. Initial program 32.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 11.3%

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

   if 1.3000000000000001e-230 < t < 5.1e14

   1. Initial program 36.6%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 7.1

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

   8. Applied rewrites 81.5%

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

   if 5.1e14 < t

   1. Initial program 36.5%

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

   3. Taylor expanded in 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 91.6

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

   5. Applied rewrites 91.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 91.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 91.6

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

   9. Applied rewrites 91.6%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-230}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\sqrt{2} \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right), \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_2 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-144)
      (/ t_2 (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
      (/ t_2 (* t_2 (sqrt (/ (+ x 1.0) (+ x -1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = t_2 / (l * sqrt(((2.0 + (2.0 / x)) / x)));
	} else {
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    if (t_m <= 3.8d-144) then
        tmp = t_2 / (l * sqrt(((2.0d0 + (2.0d0 / x)) / x)))
    else
        tmp = t_2 / (t_2 * sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = t_2 / (l * Math.sqrt(((2.0 + (2.0 / x)) / x)));
	} else {
		tmp = t_2 / (t_2 * Math.sqrt(((x + 1.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = t_m * math.sqrt(2.0)
	tmp = 0
	if t_m <= 3.8e-144:
		tmp = t_2 / (l * math.sqrt(((2.0 + (2.0 / x)) / x)))
	else:
		tmp = t_2 / (t_2 * math.sqrt(((x + 1.0) / (x + -1.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(t_2 / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	else
		tmp = Float64(t_2 / Float64(t_2 * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = t_m * sqrt(2.0);
	tmp = 0.0;
	if (t_m <= 3.8e-144)
		tmp = t_2 / (l * sqrt(((2.0 + (2.0 / x)) / x)));
	else
		tmp = t_2 / (t_2 * sqrt(((x + 1.0) / (x + -1.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(t$95$2 / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$2 * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.6%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in 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%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-144)
      (/ t_2 (* l (sqrt (/ (+ 2.0 (/ 2.0 x)) x))))
      (/ t_2 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = t_2 / (l * sqrt(((2.0 + (2.0 / x)) / x)));
	} else {
		tmp = t_2 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(t_2 / Float64(l * sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) / x))));
	else
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(t$95$2 / N[(l * N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.6%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 88.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2 + \frac{2}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-144)
      (/ t_2 (* l (sqrt (/ 2.0 x))))
      (/ t_2 (* t_m (sqrt (/ (fma x 2.0 2.0) (+ x -1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = t_2 / (l * sqrt((2.0 / x)));
	} else {
		tmp = t_2 / (t_m * sqrt((fma(x, 2.0, 2.0) / (x + -1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(t_2 / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x + -1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(t$95$2 / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 88.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-144)
    (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 x))))
    (* (sqrt 2.0) (/ t_m (* t_m (sqrt (/ (fma 2.0 x 2.0) (+ x -1.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	} else {
		tmp = sqrt(2.0) * (t_m / (t_m * sqrt((fma(2.0, x, 2.0) / (x + -1.0)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(t_m * sqrt(Float64(fma(2.0, x, 2.0) / Float64(x + -1.0))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 88.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 88.7

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

   11. Applied rewrites 88.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-144)
    (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 x))))
    (* (* (sqrt 2.0) (sqrt 0.5)) (sqrt (/ (+ x -1.0) (+ x 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	} else {
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.8d-144) then
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt((2.0d0 / x)))
    else
        tmp = (sqrt(2.0d0) * sqrt(0.5d0)) * sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt((2.0 / x)));
	} else {
		tmp = (Math.sqrt(2.0) * Math.sqrt(0.5)) * Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.8e-144:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt((2.0 / x)))
	else:
		tmp = (math.sqrt(2.0) * math.sqrt(0.5)) * math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(0.5)) * sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.8e-144)
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	else
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x + -1}{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-233}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 1.12 \cdot 10^{-179}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\left(\ell \cdot \ell\right) \cdot \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-233)
    1.0
    (if (<= t_m 1.12e-179) (* t_m (sqrt (/ 2.0 (* (* l l) (/ 2.0 x))))) 1.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4.4e-233) {
		tmp = 1.0;
	} else if (t_m <= 1.12e-179) {
		tmp = t_m * sqrt((2.0 / ((l * l) * (2.0 / x))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4.4d-233) then
        tmp = 1.0d0
    else if (t_m <= 1.12d-179) then
        tmp = t_m * sqrt((2.0d0 / ((l * l) * (2.0d0 / x))))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4.4e-233) {
		tmp = 1.0;
	} else if (t_m <= 1.12e-179) {
		tmp = t_m * Math.sqrt((2.0 / ((l * l) * (2.0 / x))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 4.4e-233:
		tmp = 1.0
	elif t_m <= 1.12e-179:
		tmp = t_m * math.sqrt((2.0 / ((l * l) * (2.0 / x))))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 4.4e-233)
		tmp = 1.0;
	elseif (t_m <= 1.12e-179)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(l * l) * Float64(2.0 / x)))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 4.4e-233)
		tmp = 1.0;
	elseif (t_m <= 1.12e-179)
		tmp = t_m * sqrt((2.0 / ((l * l) * (2.0 / x))));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-233], 1.0, If[LessEqual[t$95$m, 1.12e-179], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(l * l), $MachinePrecision] * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4e-233 or 1.11999999999999999e-179 < t

   1. Initial program 35.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 40.0%

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

   if 4.4e-233 < t < 1.11999999999999999e-179

   1. Initial program 4.9%

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

   3. Taylor expanded in l around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 4.9

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 4.7

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

   8. Applied rewrites 4.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 66.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

   13. Applied rewrites 66.4%

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

Alternative 11: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-144) (/ (* t_m (sqrt 2.0)) (* l (sqrt (/ 2.0 x)))) 1.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.8d-144) then
        tmp = (t_m * sqrt(2.0d0)) / (l * sqrt((2.0d0 / x)))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.8e-144) {
		tmp = (t_m * Math.sqrt(2.0)) / (l * Math.sqrt((2.0 / x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.8e-144:
		tmp = (t_m * math.sqrt(2.0)) / (l * math.sqrt((2.0 / x)))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.8e-144)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l * sqrt(Float64(2.0 / x))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.8e-144)
		tmp = (t_m * sqrt(2.0)) / (l * sqrt((2.0 / x)));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-144], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999993e-144

   1. Initial program 29.7%

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

   3. Taylor expanded in l around 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 2.3

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

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.5%

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

   if 3.79999999999999993e-144 < t

   1. Initial program 41.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 87.9%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.7% accurate, 85.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m) :precision binary64 (* t_s 1.0))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * 1.0;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * 1.0

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * 1.0;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 34.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 39.2

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

5. Applied rewrites 39.2%

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

7. Applied rewrites 39.8%

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

Reproduce

herbie shell --seed 2024235 
(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)))))