Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 83.6%

Time: 15.9s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 0.37:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{t\_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 (fma 2.0 (* t_m t_m) (* l l))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.2e-182)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 0.37)
        (*
         t_m
         (sqrt
          (/
           2.0
           (+
            (fma 2.0 (/ (* t_m t_m) x) (fma 2.0 (* t_m t_m) (/ (* l l) x)))
            (/ t_2 x)))))
        (/ t_3 (* 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 t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.2e-182) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 0.37) {
		tmp = t_m * sqrt((2.0 / (fma(2.0, ((t_m * t_m) / x), fma(2.0, (t_m * t_m), ((l * l) / x))) + (t_2 / x))));
	} else {
		tmp = t_3 / (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)
	t_2 = 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 <= 9.2e-182)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 0.37)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), fma(2.0, Float64(t_m * t_m), Float64(Float64(l * l) / x))) + Float64(t_2 / x)))));
	else
		tmp = Float64(t_3 / 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_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $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, 9.2e-182], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(t$95$3 * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.37], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.1999999999999996e-182

   1. Initial program 25.0%

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

   3. Taylor expanded in 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 16.0%

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

   if 9.1999999999999996e-182 < t < 0.37

   1. Initial program 57.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 57.3

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 83.1%

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

   if 0.37 < t

   1. Initial program 33.4%

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

   3. 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 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 96.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 0.37:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 0.37:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(t\_m \cdot t\_m + \frac{t\_2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 (fma 2.0 (* t_m t_m) (* l l))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.2e-182)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 0.37)
        (/ t_3 (sqrt (* 2.0 (+ (* t_m t_m) (/ t_2 x)))))
        (/ t_3 (* 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 t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.2e-182) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 0.37) {
		tmp = t_3 / sqrt((2.0 * ((t_m * t_m) + (t_2 / x))));
	} else {
		tmp = t_3 / (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)
	t_2 = 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 <= 9.2e-182)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 0.37)
		tmp = Float64(t_3 / sqrt(Float64(2.0 * Float64(Float64(t_m * t_m) + Float64(t_2 / x)))));
	else
		tmp = Float64(t_3 / 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_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $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, 9.2e-182], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(t$95$3 * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 0.37], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.1999999999999996e-182

   1. Initial program 25.0%

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

   3. Taylor expanded in 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 16.0%

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

   if 9.1999999999999996e-182 < t < 0.37

   1. Initial program 57.0%

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

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift--.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 14.4%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.9

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

   7. Applied rewrites 82.9%

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

   if 0.37 < t

   1. Initial program 33.4%

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

   3. 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 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 96.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-182}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 0.37:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.37:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot \left(t\_m \cdot t\_m + \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 0.37)
      (/
       t_2
       (sqrt (* 2.0 (+ (* t_m t_m) (/ (fma 2.0 (* t_m t_m) (* l l)) x)))))
      (/ t_2 (* 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 t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 0.37) {
		tmp = t_2 / sqrt((2.0 * ((t_m * t_m) + (fma(2.0, (t_m * t_m), (l * l)) / x))));
	} else {
		tmp = t_2 / (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)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 0.37)
		tmp = Float64(t_2 / sqrt(Float64(2.0 * Float64(Float64(t_m * t_m) + Float64(fma(2.0, Float64(t_m * t_m), Float64(l * l)) / x)))));
	else
		tmp = Float64(t_2 / 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_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 0.37], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.37

   1. Initial program 30.1%

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

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift--.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 13.8%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

   if 0.37 < t

   1. Initial program 33.4%

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

   3. 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 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 96.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.37:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

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 9e-236)
      (/ t_2 (sqrt (/ (fma l l (* l l)) x)))
      (/ t_2 (* 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 t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9e-236) {
		tmp = t_2 / sqrt((fma(l, l, (l * l)) / x));
	} else {
		tmp = t_2 / (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)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9e-236)
		tmp = Float64(t_2 / sqrt(Float64(fma(l, l, Float64(l * l)) / x)));
	else
		tmp = Float64(t_2 / 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_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-236], N[(t$95$2 / N[Sqrt[N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999997e-236

   1. Initial program 26.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.4%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   9. Applied rewrites 84.8%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{t\_m \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \end{array} \]

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 9e-236)
    (/ (* t_m (sqrt 2.0)) (sqrt (/ (fma l l (* l l)) x)))
    (* t_m (/ (sqrt 2.0) (* 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 <= 9e-236) {
		tmp = (t_m * sqrt(2.0)) / sqrt((fma(l, l, (l * l)) / x));
	} else {
		tmp = t_m * (sqrt(2.0) / (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 <= 9e-236)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(fma(l, l, Float64(l * l)) / x)));
	else
		tmp = Float64(t_m * Float64(sqrt(2.0) / 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, 9e-236], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / 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 < 8.99999999999999997e-236

   1. Initial program 26.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.4%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.5

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

   9. Applied rewrites 84.5%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{\frac{\mathsf{fma}\left(2, x, 2\right)}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_m \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \end{array} \]

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 9e-236)
      (/ t_2 (sqrt (/ (fma l l (* l l)) x)))
      (/ t_2 (* t_m (sqrt (+ 2.0 (/ 4.0 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 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9e-236) {
		tmp = t_2 / sqrt((fma(l, l, (l * l)) / x));
	} else {
		tmp = t_2 / (t_m * sqrt((2.0 + (4.0 / 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(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9e-236)
		tmp = Float64(t_2 / sqrt(Float64(fma(l, l, Float64(l * l)) / x)));
	else
		tmp = Float64(t_2 / Float64(t_m * sqrt(Float64(2.0 + Float64(4.0 / 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 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9e-236], N[(t$95$2 / N[Sqrt[N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$m * N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999997e-236

   1. Initial program 26.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 3.4

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

   5. Applied rewrites 3.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.4%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 83.3%

       \[\leadsto \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 + \frac{4}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\left(\ell \cdot \ell\right) \cdot \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{t\_m \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

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 9e-236)
    (* t_m (sqrt (/ 2.0 (* (* l l) (/ 2.0 x)))))
    (/ (* t_m (sqrt 2.0)) (* t_m (sqrt (+ 2.0 (/ 4.0 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 tmp;
	if (t_m <= 9e-236) {
		tmp = t_m * sqrt((2.0 / ((l * l) * (2.0 / x))));
	} else {
		tmp = (t_m * sqrt(2.0)) / (t_m * sqrt((2.0 + (4.0 / x))));
	}
	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 <= 9d-236) then
        tmp = t_m * sqrt((2.0d0 / ((l * l) * (2.0d0 / x))))
    else
        tmp = (t_m * sqrt(2.0d0)) / (t_m * sqrt((2.0d0 + (4.0d0 / x))))
    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 <= 9e-236) {
		tmp = t_m * Math.sqrt((2.0 / ((l * l) * (2.0 / x))));
	} else {
		tmp = (t_m * Math.sqrt(2.0)) / (t_m * Math.sqrt((2.0 + (4.0 / x))));
	}
	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 <= 9e-236:
		tmp = t_m * math.sqrt((2.0 / ((l * l) * (2.0 / x))))
	else:
		tmp = (t_m * math.sqrt(2.0)) / (t_m * math.sqrt((2.0 + (4.0 / x))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 9e-236)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(l * l) * Float64(2.0 / x)))));
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(t_m * sqrt(Float64(2.0 + Float64(4.0 / x)))));
	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 <= 9e-236)
		tmp = t_m * sqrt((2.0 / ((l * l) * (2.0 / x))));
	else
		tmp = (t_m * sqrt(2.0)) / (t_m * sqrt((2.0 + (4.0 / x))));
	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, 9e-236], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(l * l), $MachinePrecision] * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999997e-236

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 26.8

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 3.5

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 3.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 21.5%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 83.3%

       \[\leadsto \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 + \frac{4}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-236}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\left(\ell \cdot \ell\right) \cdot \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\left(\ell \cdot \ell\right) \cdot \frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 9e-236) (* 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 <= 9e-236) {
		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 <= 9d-236) 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 <= 9e-236) {
		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 <= 9e-236:
		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 <= 9e-236)
		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 <= 9e-236)
		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, 9e-236], 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 < 8.99999999999999997e-236

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 26.8

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 3.5

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 3.5%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 21.5%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 83.2%

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

Alternative 9: 77.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-236}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 9e-236) (* t_m (sqrt (/ 2.0 (/ (* 2.0 (* l l)) 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 <= 9e-236) {
		tmp = t_m * sqrt((2.0 / ((2.0 * (l * l)) / 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 <= 9d-236) then
        tmp = t_m * sqrt((2.0d0 / ((2.0d0 * (l * l)) / 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 <= 9e-236) {
		tmp = t_m * Math.sqrt((2.0 / ((2.0 * (l * l)) / 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 <= 9e-236:
		tmp = t_m * math.sqrt((2.0 / ((2.0 * (l * l)) / 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 <= 9e-236)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(l * l)) / 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 <= 9e-236)
		tmp = t_m * sqrt((2.0 / ((2.0 * (l * l)) / 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, 9e-236], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999997e-236

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 26.8

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 3.5

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

   7. Applied rewrites 3.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 21.5%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 21.5%

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

   if 8.99999999999999997e-236 < t

   1. Initial program 36.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 82.0

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 83.2%

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

Alternative 10: 75.6% 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 31.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 40.7

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

5. Applied rewrites 40.7%

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

7. Applied rewrites 41.3%

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

Reproduce

herbie shell --seed 2024233 
(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)))))