Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 81.9%

Time: 11.0s

Alternatives: 5

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: 33.4% 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: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \frac{t\_2}{\sqrt{\left(\left(t\_m \cdot t\_m\right) \cdot 2 + l\_m \cdot l\_m\right) \cdot \frac{x - -1}{x - 1} - l\_m \cdot l\_m}}\\ t_4 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(t\_m \cdot 2, t\_m, \frac{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{x}, 2, \frac{l\_m \cdot l\_m}{x}\right) + \mathsf{fma}\left(2, t\_4, \frac{t\_4}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\_m\right) \cdot \sqrt{2}\right)}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3
         (/
          t_2
          (sqrt
           (-
            (* (+ (* (* t_m t_m) 2.0) (* l_m l_m)) (/ (- x -1.0) (- x 1.0)))
            (* l_m l_m)))))
        (t_4 (fma (* t_m t_m) 2.0 (* l_m l_m))))
   (*
    t_s
    (if (<= t_3 0.0)
      (sqrt (/ (- x 1.0) (- x -1.0)))
      (if (<= t_3 INFINITY)
        (/
         t_2
         (sqrt
          (fma
           (* t_m 2.0)
           t_m
           (/
            (+
             (fma (/ (* t_m t_m) x) 2.0 (/ (* l_m l_m) x))
             (fma 2.0 t_4 (/ t_4 x)))
            x))))
        (/ (* (sqrt x) (* (* (sqrt 0.5) t_m) (sqrt 2.0))) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = t_2 / sqrt((((((t_m * t_m) * 2.0) + (l_m * l_m)) * ((x - -1.0) / (x - 1.0))) - (l_m * l_m)));
	double t_4 = fma((t_m * t_m), 2.0, (l_m * l_m));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2 / sqrt(fma((t_m * 2.0), t_m, ((fma(((t_m * t_m) / x), 2.0, ((l_m * l_m) / x)) + fma(2.0, t_4, (t_4 / x))) / x)));
	} else {
		tmp = (sqrt(x) * ((sqrt(0.5) * t_m) * sqrt(2.0))) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(t_2 / sqrt(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 2.0) + Float64(l_m * l_m)) * Float64(Float64(x - -1.0) / Float64(x - 1.0))) - Float64(l_m * l_m))))
	t_4 = fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_2 / sqrt(fma(Float64(t_m * 2.0), t_m, Float64(Float64(fma(Float64(Float64(t_m * t_m) / x), 2.0, Float64(Float64(l_m * l_m) / x)) + fma(2.0, t_4, Float64(t_4 / x))) / x))));
	else
		tmp = Float64(Float64(sqrt(x) * Float64(Float64(sqrt(0.5) * t_m) * sqrt(2.0))) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 0.0], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$2 / N[Sqrt[N[(N[(t$95$m * 2.0), $MachinePrecision] * t$95$m + N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$4 + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 0.0

   1. Initial program 34.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 18.2

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

   5. Applied rewrites 18.2%

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

   7. Applied rewrites 18.4%

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

   if 0.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 55.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 86.9%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.7%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} - \ell \cdot \ell}} \leq 0:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(t \cdot 2, t, \frac{\mathsf{fma}\left(\frac{t \cdot t}{x}, 2, \frac{\ell \cdot \ell}{x}\right) + \mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\right) \cdot \sqrt{2}\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \frac{t\_2}{\sqrt{\left(\left(t\_m \cdot t\_m\right) \cdot 2 + l\_m \cdot l\_m\right) \cdot \frac{x - -1}{x - 1} - l\_m \cdot l\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 1.000000000005:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x} + t\_m \cdot t\_m, \frac{l\_m \cdot l\_m}{x} + \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, l\_m \cdot l\_m\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\_m\right) \cdot \sqrt{2}\right)}{l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3
         (/
          t_2
          (sqrt
           (-
            (* (+ (* (* t_m t_m) 2.0) (* l_m l_m)) (/ (- x -1.0) (- x 1.0)))
            (* l_m l_m))))))
   (*
    t_s
    (if (<= t_3 1.000000000005)
      (sqrt (/ (- x 1.0) (- x -1.0)))
      (if (<= t_3 INFINITY)
        (/
         t_2
         (sqrt
          (fma
           2.0
           (+ (/ (* t_m t_m) x) (* t_m t_m))
           (+ (/ (* l_m l_m) x) (/ (fma (* t_m t_m) 2.0 (* l_m l_m)) x)))))
        (/ (* (sqrt x) (* (* (sqrt 0.5) t_m) (sqrt 2.0))) l_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = t_2 / sqrt((((((t_m * t_m) * 2.0) + (l_m * l_m)) * ((x - -1.0) / (x - 1.0))) - (l_m * l_m)));
	double tmp;
	if (t_3 <= 1.000000000005) {
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2 / sqrt(fma(2.0, (((t_m * t_m) / x) + (t_m * t_m)), (((l_m * l_m) / x) + (fma((t_m * t_m), 2.0, (l_m * l_m)) / x))));
	} else {
		tmp = (sqrt(x) * ((sqrt(0.5) * t_m) * sqrt(2.0))) / l_m;
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(t_2 / sqrt(Float64(Float64(Float64(Float64(Float64(t_m * t_m) * 2.0) + Float64(l_m * l_m)) * Float64(Float64(x - -1.0) / Float64(x - 1.0))) - Float64(l_m * l_m))))
	tmp = 0.0
	if (t_3 <= 1.000000000005)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(Float64(Float64(t_m * t_m) / x) + Float64(t_m * t_m)), Float64(Float64(Float64(l_m * l_m) / x) + Float64(fma(Float64(t_m * t_m), 2.0, Float64(l_m * l_m)) / x)))));
	else
		tmp = Float64(Float64(sqrt(x) * Float64(Float64(sqrt(0.5) * t_m) * sqrt(2.0))) / l_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 1.000000000005], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < 1.000000000005

   1. Initial program 47.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 33.6%

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

   if 1.000000000005 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 2.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]
   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. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

   5. Applied rewrites 72.9%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x #s(literal 1 binary64)) (-.f64 x #s(literal 1 binary64))) (+.f64 (*.f64 l l) (*.f64 #s(literal 2 binary64) (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.7%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} - \ell \cdot \ell}} \leq 1.000000000005:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{elif}\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\left(t \cdot t\right) \cdot 2 + \ell \cdot \ell\right) \cdot \frac{x - -1}{x - 1} - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x} + t \cdot t, \frac{\ell \cdot \ell}{x} + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\right) \cdot \sqrt{2}\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.9 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\_m\right) \cdot \sqrt{2}\right)}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.9e+125)
    (sqrt (/ (- x 1.0) (- x -1.0)))
    (/ (* (sqrt x) (* (* (sqrt 0.5) t_m) (sqrt 2.0))) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.9e+125) {
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	} else {
		tmp = (sqrt(x) * ((sqrt(0.5) * t_m) * sqrt(2.0))) / l_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.9d+125) then
        tmp = sqrt(((x - 1.0d0) / (x - (-1.0d0))))
    else
        tmp = (sqrt(x) * ((sqrt(0.5d0) * t_m) * sqrt(2.0d0))) / l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.9e+125) {
		tmp = Math.sqrt(((x - 1.0) / (x - -1.0)));
	} else {
		tmp = (Math.sqrt(x) * ((Math.sqrt(0.5) * t_m) * Math.sqrt(2.0))) / l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.9e+125:
		tmp = math.sqrt(((x - 1.0) / (x - -1.0)))
	else:
		tmp = (math.sqrt(x) * ((math.sqrt(0.5) * t_m) * math.sqrt(2.0))) / l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.9e+125)
		tmp = sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0)));
	else
		tmp = Float64(Float64(sqrt(x) * Float64(Float64(sqrt(0.5) * t_m) * sqrt(2.0))) / l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.9e+125)
		tmp = sqrt(((x - 1.0) / (x - -1.0)));
	else
		tmp = (sqrt(x) * ((sqrt(0.5) * t_m) * sqrt(2.0))) / l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.9e+125], N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.89999999999999993e125

   1. Initial program 38.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 37.3

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

   5. Applied rewrites 37.3%

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

   7. Applied rewrites 37.9%

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

   if 2.89999999999999993e125 < l

   1. Initial program 0.8%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.5%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot \left(\left(\sqrt{0.5} \cdot t\right) \cdot \sqrt{2}\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \sqrt{\frac{x - 1}{x - -1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in t around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 33.9

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

5. Applied rewrites 33.9%

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

7. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{x - -1}}} \]
8. Add Preprocessing

Alternative 5: 76.0% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 33.6

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

5. Applied rewrites 33.6%

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

7. Applied rewrites 34.1%

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

Reproduce

herbie shell --seed 2024248 
(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)))))