Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 85.3%

Time: 19.1s

Alternatives: 9

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% 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: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m, t\_m, t\_m \cdot t\_m\right)\\ t_3 := \mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)\\ t_4 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{t\_4 \cdot x}, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_4}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) - \frac{\left(-2 \cdot t\_2 - t\_3\right) - \frac{\mathsf{fma}\left(-2, -t\_2, t\_3\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{t\_m \cdot \left(\sqrt{2} + \frac{2}{\sqrt{2} \cdot x}\right)}\\ \end{array} \end{array} \end{array} \]

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 (fma t_m t_m (* t_m t_m)))
        (t_3 (fma l l (* l l)))
        (t_4 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-160)
      (/ t_4 (fma 0.5 (/ (* 2.0 (fma 2.0 (* t_m t_m) (* l l))) (* t_4 x)) t_4))
      (if (<= t_m 2.85e+57)
        (/
         t_4
         (sqrt
          (-
           (* 2.0 (* t_m t_m))
           (/ (- (- (* -2.0 t_2) t_3) (/ (fma -2.0 (- t_2) t_3) x)) x))))
        (/ t_4 (* t_m (+ (sqrt 2.0) (/ 2.0 (* (sqrt 2.0) x))))))))))

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 = fma(t_m, t_m, (t_m * t_m));
	double t_3 = fma(l, l, (l * l));
	double t_4 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-160) {
		tmp = t_4 / fma(0.5, ((2.0 * fma(2.0, (t_m * t_m), (l * l))) / (t_4 * x)), t_4);
	} else if (t_m <= 2.85e+57) {
		tmp = t_4 / sqrt(((2.0 * (t_m * t_m)) - ((((-2.0 * t_2) - t_3) - (fma(-2.0, -t_2, t_3) / x)) / x)));
	} else {
		tmp = t_4 / (t_m * (sqrt(2.0) + (2.0 / (sqrt(2.0) * x))));
	}
	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 = fma(t_m, t_m, Float64(t_m * t_m))
	t_3 = fma(l, l, Float64(l * l))
	t_4 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-160)
		tmp = Float64(t_4 / fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t_m * t_m), Float64(l * l))) / Float64(t_4 * x)), t_4));
	elseif (t_m <= 2.85e+57)
		tmp = Float64(t_4 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) - Float64(Float64(Float64(Float64(-2.0 * t_2) - t_3) - Float64(fma(-2.0, Float64(-t_2), t_3) / x)) / x))));
	else
		tmp = Float64(t_4 / Float64(t_m * Float64(sqrt(2.0) + Float64(2.0 / Float64(sqrt(2.0) * x)))));
	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 * t$95$m + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-160], N[(t$95$4 / N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * x), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+57], N[(t$95$4 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-2.0 * t$95$2), $MachinePrecision] - t$95$3), $MachinePrecision] - N[(N[(-2.0 * (-t$95$2) + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.5000000000000002e-160

   1. Initial program 3.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{1}{2} \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) - -1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}} \]
   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 66.9%

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

   if 9.5000000000000002e-160 < t < 2.8499999999999999e57

   1. Initial program 54.1%

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

      1. lift--.f64 N/A

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

      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. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. distribute-rgt-in N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r/ N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 49.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lift--.f64 N/A

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

      10. associate-/r/ N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in x around -inf

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

   9. Applied rewrites 88.1%

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

   if 2.8499999999999999e57 < t

   1. Initial program 24.3%

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

   3. Taylor expanded in 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 26.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 92.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) - \frac{\left(-2 \cdot \mathsf{fma}\left(t, t, t \cdot t\right) - \mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)\right) - \frac{\mathsf{fma}\left(-2, -\mathsf{fma}\left(t, t, t \cdot t\right), \mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)\right)}{x}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{2}{\sqrt{2} \cdot x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{t\_2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \left(\sqrt{2} + \frac{2}{\sqrt{2} \cdot x}\right)}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (fma 2.0 (* t_m t_m) (* l l)))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-160)
      (/ t_3 (fma 0.5 (/ t_2 (* t_3 x)) t_3))
      (if (<= t_m 2.85e+57)
        (/ t_3 (sqrt (fma 2.0 (* t_m t_m) (/ t_2 x))))
        (/ t_3 (* t_m (+ (sqrt 2.0) (/ 2.0 (* (sqrt 2.0) x))))))))))

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 = 2.0 * fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-160) {
		tmp = t_3 / fma(0.5, (t_2 / (t_3 * x)), t_3);
	} else if (t_m <= 2.85e+57) {
		tmp = t_3 / sqrt(fma(2.0, (t_m * t_m), (t_2 / x)));
	} else {
		tmp = t_3 / (t_m * (sqrt(2.0) + (2.0 / (sqrt(2.0) * x))));
	}
	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(2.0 * fma(2.0, Float64(t_m * t_m), Float64(l * l)))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-160)
		tmp = Float64(t_3 / fma(0.5, Float64(t_2 / Float64(t_3 * x)), t_3));
	elseif (t_m <= 2.85e+57)
		tmp = Float64(t_3 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(t_2 / x))));
	else
		tmp = Float64(t_3 / Float64(t_m * Float64(sqrt(2.0) + Float64(2.0 / Float64(sqrt(2.0) * x)))));
	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[(2.0 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-160], N[(t$95$3 / N[(0.5 * N[(t$95$2 / N[(t$95$3 * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+57], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.5000000000000002e-160

   1. Initial program 3.7%

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

   3. Taylor expanded in 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 61.7%

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

   if 9.5000000000000002e-160 < t < 2.8499999999999999e57

   1. Initial program 56.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   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 84.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 84.9%

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

   if 2.8499999999999999e57 < t

   1. Initial program 29.9%

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

   3. Taylor expanded in 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 30.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 94.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(\sqrt{2} + \frac{2}{\sqrt{2} \cdot x}\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))))