Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 99.0%

Time: 25.0s

Alternatives: 11

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

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

FPCore

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

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 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = pow(t_2, -0.5);
	} else {
		tmp = sqrt(2.0) * (t_m / (sqrt(2.0) * hypot((l / sqrt(x)), (t_m * sqrt(t_2)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = (x + 1.0) / (x + -1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = Math.pow(t_2, -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (Math.sqrt(2.0) * Math.hypot((l / Math.sqrt(x)), (t_m * Math.sqrt(t_2)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = (x + 1.0) / (x + -1.0)
	tmp = 0
	if x <= -1.0:
		tmp = math.pow(t_2, -0.5)
	else:
		tmp = math.sqrt(2.0) * (t_m / (math.sqrt(2.0) * math.hypot((l / math.sqrt(x)), (t_m * math.sqrt(t_2)))))
	return t_s * tmp

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = (x + 1.0) / (x + -1.0);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_2 ^ -0.5;
	else
		tmp = sqrt(2.0) * (t_m / (sqrt(2.0) * hypot((l / sqrt(x)), (t_m * sqrt(t_2)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 41.8%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in t around inf 45.7%

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

   6. Taylor expanded in t around 0 45.8%

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

      1. clear-num 45.8%

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

      2. +-commutative 45.8%

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

      3. sub-neg 45.8%

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

      4. metadata-eval 45.8%

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

      5. +-commutative 45.8%

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

      6. inv-pow 45.8%

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

      7. +-commutative 45.8%

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

   8. Applied egg-rr 45.8%

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

      1. unpow-1 45.8%

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

      2. +-commutative 45.8%

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

   10. Simplified 45.8%

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

       1. *-un-lft-identity 45.8%

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

       2. inv-pow 45.8%

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

       3. sqrt-pow1 45.8%

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

       4. +-commutative 45.8%

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

       5. +-commutative 45.8%

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

       6. metadata-eval 45.8%

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

   12. Applied egg-rr 45.8%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 45.8%

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

       2. +-commutative 45.8%

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

       3. +-commutative 45.8%

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

   14. Simplified 45.8%

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

   if -1 < x

   1. Initial program 32.7%

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

   3. Simplified 27.6%

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

   5. Taylor expanded in l around 0 27.1%

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

      1. fma-define 27.1%

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

      2. sub-neg 27.1%

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

      3. metadata-eval 27.1%

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

      4. associate-/l* 39.7%

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

      5. +-commutative 39.7%

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

      6. +-commutative 39.7%

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

      7. associate--l+ 48.1%

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

      8. sub-neg 48.1%

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

      9. metadata-eval 48.1%

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

      10. +-commutative 48.1%

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

      11. sub-neg 48.1%

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

      12. metadata-eval 48.1%

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

      13. +-commutative 48.1%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in x around inf 57.4%

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

   9. Taylor expanded in t around 0 44.8%

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

       1. distribute-lft-out 44.8%

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

       2. sub-neg 44.8%

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

       3. metadata-eval 44.8%

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

       4. associate-/l* 57.4%

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

       5. +-commutative 57.4%

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

       6. +-commutative 57.4%

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

   11. Simplified 57.4%

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

       1. *-commutative 57.4%

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

       2. sqrt-prod 57.6%

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

   13. Applied egg-rr 99.7%

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

4. Final simplification 78.0%

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

Alternative 2: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x + 1}{x + -1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+138}:\\ \;\;\;\;{t\_2}^{-0.5}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+269}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{2 \cdot \left(t\_2 \cdot \left(t\_m \cdot t\_m\right) + \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \end{array} \]

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 (/ (+ x 1.0) (+ x -1.0))))
   (*
    t_s
    (if (<= (* l l) 5e+138)
      (pow t_2 -0.5)
      (if (<= (* l l) 5e+269)
        (*
         (sqrt 2.0)
         (/ t_m (sqrt (* 2.0 (+ (* t_2 (* t_m t_m)) (/ (pow l 2.0) x))))))
        (* (sqrt 2.0) (/ t_m (* (* (sqrt 2.0) l) (sqrt (/ 1.0 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 = (x + 1.0) / (x + -1.0);
	double tmp;
	if ((l * l) <= 5e+138) {
		tmp = pow(t_2, -0.5);
	} else if ((l * l) <= 5e+269) {
		tmp = sqrt(2.0) * (t_m / sqrt((2.0 * ((t_2 * (t_m * t_m)) + (pow(l, 2.0) / x)))));
	} else {
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * sqrt((1.0 / 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) :: t_2
    real(8) :: tmp
    t_2 = (x + 1.0d0) / (x + (-1.0d0))
    if ((l * l) <= 5d+138) then
        tmp = t_2 ** (-0.5d0)
    else if ((l * l) <= 5d+269) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((2.0d0 * ((t_2 * (t_m * t_m)) + ((l ** 2.0d0) / x)))))
    else
        tmp = sqrt(2.0d0) * (t_m / ((sqrt(2.0d0) * l) * sqrt((1.0d0 / 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 t_2 = (x + 1.0) / (x + -1.0);
	double tmp;
	if ((l * l) <= 5e+138) {
		tmp = Math.pow(t_2, -0.5);
	} else if ((l * l) <= 5e+269) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((2.0 * ((t_2 * (t_m * t_m)) + (Math.pow(l, 2.0) / x)))));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / ((Math.sqrt(2.0) * l) * Math.sqrt((1.0 / 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):
	t_2 = (x + 1.0) / (x + -1.0)
	tmp = 0
	if (l * l) <= 5e+138:
		tmp = math.pow(t_2, -0.5)
	elif (l * l) <= 5e+269:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((2.0 * ((t_2 * (t_m * t_m)) + (math.pow(l, 2.0) / x)))))
	else:
		tmp = math.sqrt(2.0) * (t_m / ((math.sqrt(2.0) * l) * math.sqrt((1.0 / 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(Float64(x + 1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (Float64(l * l) <= 5e+138)
		tmp = t_2 ^ -0.5;
	elseif (Float64(l * l) <= 5e+269)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(2.0 * Float64(Float64(t_2 * Float64(t_m * t_m)) + Float64((l ^ 2.0) / x))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / 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)
	t_2 = (x + 1.0) / (x + -1.0);
	tmp = 0.0;
	if ((l * l) <= 5e+138)
		tmp = t_2 ^ -0.5;
	elseif ((l * l) <= 5e+269)
		tmp = sqrt(2.0) * (t_m / sqrt((2.0 * ((t_2 * (t_m * t_m)) + ((l ^ 2.0) / x)))));
	else
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * sqrt((1.0 / 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_] := Block[{t$95$2 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+138], N[Power[t$95$2, -0.5], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+269], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.00000000000000016e138

   1. Initial program 47.8%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around inf 44.5%

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

   6. Taylor expanded in t around 0 44.6%

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

      1. clear-num 44.6%

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

      2. +-commutative 44.6%

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

      3. sub-neg 44.6%

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

      4. metadata-eval 44.6%

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

      5. +-commutative 44.6%

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

      6. inv-pow 44.6%

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

      7. +-commutative 44.6%

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

   8. Applied egg-rr 44.6%

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

      1. unpow-1 44.6%

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

      2. +-commutative 44.6%

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

   10. Simplified 44.6%

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

       1. *-un-lft-identity 44.6%

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

       2. inv-pow 44.6%

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

       3. sqrt-pow1 44.6%

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

       4. +-commutative 44.6%

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

       5. +-commutative 44.6%

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

       6. metadata-eval 44.6%

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

   12. Applied egg-rr 44.6%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 44.6%

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

       2. +-commutative 44.6%

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

       3. +-commutative 44.6%

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

   14. Simplified 44.6%

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

   if 5.00000000000000016e138 < (*.f64 l l) < 5.0000000000000002e269

   1. Initial program 19.3%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in l around 0 23.7%

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

      1. fma-define 23.7%

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

      2. sub-neg 23.7%

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

      3. metadata-eval 23.7%

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

      4. associate-/l* 44.5%

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

      5. +-commutative 44.5%

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

      6. +-commutative 44.5%

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

      7. associate--l+ 62.6%

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

      8. sub-neg 62.6%

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

      9. metadata-eval 62.6%

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

      10. +-commutative 62.6%

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

      11. sub-neg 62.6%

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

      12. metadata-eval 62.6%

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

      13. +-commutative 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in x around inf 86.6%

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

   9. Taylor expanded in t around 0 65.7%

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

       1. distribute-lft-out 65.7%

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

       2. sub-neg 65.7%

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

       3. metadata-eval 65.7%

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

       4. associate-/l* 86.6%

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

       5. +-commutative 86.6%

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

       6. +-commutative 86.6%

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

   11. Simplified 86.6%

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

       1. unpow2 86.6%

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

   13. Applied egg-rr 86.6%

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

   if 5.0000000000000002e269 < (*.f64 l l)

   1. Initial program 0.2%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in l around inf 1.8%

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

      1. associate--l+ 19.2%

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

      2. sub-neg 19.2%

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

      3. metadata-eval 19.2%

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

      4. +-commutative 19.2%

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

      5. sub-neg 19.2%

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

      6. metadata-eval 19.2%

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

      7. +-commutative 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in x around inf 39.3%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+138}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+269}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{x + 1}{x + -1} \cdot \left(t \cdot t\right) + \frac{{\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

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 (<= l 7.2e+167)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (* (sqrt 2.0) (/ t_m (* (* (sqrt 2.0) l) (sqrt (/ 1.0 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 (l <= 7.2e+167) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * sqrt((1.0 / 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 (l <= 7.2d+167) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else
        tmp = sqrt(2.0d0) * (t_m / ((sqrt(2.0d0) * l) * sqrt((1.0d0 / 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 (l <= 7.2e+167) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (t_m / ((Math.sqrt(2.0) * l) * Math.sqrt((1.0 / 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 l <= 7.2e+167:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	else:
		tmp = math.sqrt(2.0) * (t_m / ((math.sqrt(2.0) * l) * math.sqrt((1.0 / 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 (l <= 7.2e+167)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / 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 (l <= 7.2e+167)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	else
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * sqrt((1.0 / 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[l, 7.2e+167], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000049e167

   1. Initial program 39.6%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in t around inf 41.0%

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

   6. Taylor expanded in t around 0 41.1%

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

      1. clear-num 41.1%

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

      2. +-commutative 41.1%

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

      3. sub-neg 41.1%

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

      4. metadata-eval 41.1%

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

      5. +-commutative 41.1%

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

      6. inv-pow 41.1%

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

      7. +-commutative 41.1%

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

   8. Applied egg-rr 41.1%

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

      1. unpow-1 41.1%

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

      2. +-commutative 41.1%

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

   10. Simplified 41.1%

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

       1. *-un-lft-identity 41.1%

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

       2. inv-pow 41.1%

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

       3. sqrt-pow1 41.1%

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

       4. +-commutative 41.1%

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

       5. +-commutative 41.1%

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

       6. metadata-eval 41.1%

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

   12. Applied egg-rr 41.1%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. +-commutative 41.1%

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

       3. +-commutative 41.1%

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

   14. Simplified 41.1%

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

   if 7.20000000000000049e167 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 2.5%

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

      1. associate--l+ 21.7%

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

      2. sub-neg 21.7%

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

      3. metadata-eval 21.7%

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

      4. +-commutative 21.7%

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

      5. sub-neg 21.7%

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

      6. metadata-eval 21.7%

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

      7. +-commutative 21.7%

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

   7. Simplified 21.7%

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

   8. Taylor expanded in x around inf 63.0%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \end{array} \end{array} \]

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 (<= l 5e+167)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (* (sqrt 2.0) (/ t_m (* l (* (sqrt 2.0) (sqrt (/ 1.0 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 (l <= 5e+167) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = sqrt(2.0) * (t_m / (l * (sqrt(2.0) * sqrt((1.0 / 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 (l <= 5d+167) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else
        tmp = sqrt(2.0d0) * (t_m / (l * (sqrt(2.0d0) * sqrt((1.0d0 / 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 (l <= 5e+167) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / 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 l <= 5e+167:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * (math.sqrt(2.0) * math.sqrt((1.0 / 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 (l <= 5e+167)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / 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 (l <= 5e+167)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	else
		tmp = sqrt(2.0) * (t_m / (l * (sqrt(2.0) * sqrt((1.0 / 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[l, 5e+167], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999997e167

   1. Initial program 39.6%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in t around inf 41.0%

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

   6. Taylor expanded in t around 0 41.1%

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

      1. clear-num 41.1%

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

      2. +-commutative 41.1%

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

      3. sub-neg 41.1%

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

      4. metadata-eval 41.1%

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

      5. +-commutative 41.1%

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

      6. inv-pow 41.1%

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

      7. +-commutative 41.1%

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

   8. Applied egg-rr 41.1%

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

      1. unpow-1 41.1%

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

      2. +-commutative 41.1%

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

   10. Simplified 41.1%

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

       1. *-un-lft-identity 41.1%

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

       2. inv-pow 41.1%

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

       3. sqrt-pow1 41.1%

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

       4. +-commutative 41.1%

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

       5. +-commutative 41.1%

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

       6. metadata-eval 41.1%

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

   12. Applied egg-rr 41.1%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. +-commutative 41.1%

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

       3. +-commutative 41.1%

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

   14. Simplified 41.1%

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

   if 4.9999999999999997e167 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 2.5%

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

      1. associate--l+ 21.7%

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

      2. sub-neg 21.7%

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

      3. metadata-eval 21.7%

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

      4. +-commutative 21.7%

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

      5. sub-neg 21.7%

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

      6. metadata-eval 21.7%

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

      7. +-commutative 21.7%

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

   7. Simplified 21.7%

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

   8. Taylor expanded in x around inf 63.0%

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

      1. associate-*l* 63.0%

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

      2. *-commutative 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 5e+167)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (* (sqrt 2.0) (/ t_m (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 5e+167) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 5d+167) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 5e+167) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 5e+167:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 5e+167)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 5e+167)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 5e+167], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999997e167

   1. Initial program 39.6%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in t around inf 41.0%

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

   6. Taylor expanded in t around 0 41.1%

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

      1. clear-num 41.1%

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

      2. +-commutative 41.1%

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

      3. sub-neg 41.1%

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

      4. metadata-eval 41.1%

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

      5. +-commutative 41.1%

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

      6. inv-pow 41.1%

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

      7. +-commutative 41.1%

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

   8. Applied egg-rr 41.1%

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

      1. unpow-1 41.1%

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

      2. +-commutative 41.1%

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

   10. Simplified 41.1%

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

       1. *-un-lft-identity 41.1%

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

       2. inv-pow 41.1%

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

       3. sqrt-pow1 41.1%

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

       4. +-commutative 41.1%

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

       5. +-commutative 41.1%

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

       6. metadata-eval 41.1%

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

   12. Applied egg-rr 41.1%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 41.1%

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

       2. +-commutative 41.1%

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

       3. +-commutative 41.1%

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

   14. Simplified 41.1%

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

   if 4.9999999999999997e167 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 2.5%

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

      1. associate--l+ 21.7%

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

      2. sub-neg 21.7%

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

      3. metadata-eval 21.7%

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

      4. +-commutative 21.7%

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

      5. sub-neg 21.7%

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

      6. metadata-eval 21.7%

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

      7. +-commutative 21.7%

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

   7. Simplified 21.7%

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

   8. Taylor expanded in x around inf 62.8%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+209}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \end{array} \end{array} \]

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 (<= l 6.5e+209)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (* (sqrt x) (/ t_m l)))))

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 (l <= 6.5e+209) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = sqrt(x) * (t_m / l);
	}
	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 (l <= 6.5d+209) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else
        tmp = sqrt(x) * (t_m / l)
    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 (l <= 6.5e+209) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = Math.sqrt(x) * (t_m / l);
	}
	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 l <= 6.5e+209:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	else:
		tmp = math.sqrt(x) * (t_m / l)
	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 (l <= 6.5e+209)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	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 (l <= 6.5e+209)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	else
		tmp = sqrt(x) * (t_m / l);
	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[l, 6.5e+209], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.49999999999999975e209

   1. Initial program 38.3%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in t around inf 40.6%

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

   6. Taylor expanded in t around 0 40.6%

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

      1. clear-num 40.6%

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

      2. +-commutative 40.6%

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

      3. sub-neg 40.6%

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

      4. metadata-eval 40.6%

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

      5. +-commutative 40.6%

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

      6. inv-pow 40.6%

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

      7. +-commutative 40.6%

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

   8. Applied egg-rr 40.6%

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

      1. unpow-1 40.6%

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

      2. +-commutative 40.6%

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

   10. Simplified 40.6%

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

       1. *-un-lft-identity 40.6%

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

       2. inv-pow 40.6%

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

       3. sqrt-pow1 40.6%

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

       4. +-commutative 40.6%

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

       5. +-commutative 40.6%

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

       6. metadata-eval 40.6%

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

   12. Applied egg-rr 40.6%

       \[\leadsto \color{blue}{1 \cdot {\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
   13. Step-by-step derivation

       1. *-lft-identity 40.6%

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

       2. +-commutative 40.6%

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

       3. +-commutative 40.6%

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

   14. Simplified 40.6%

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

   if 6.49999999999999975e209 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around 0 0.0%

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

      1. fma-define 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. associate-/l* 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. associate--l+ 18.7%

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

      8. sub-neg 18.7%

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

      9. metadata-eval 18.7%

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

      10. +-commutative 18.7%

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

      11. sub-neg 18.7%

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

      12. metadata-eval 18.7%

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

      13. +-commutative 18.7%

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

   7. Simplified 18.7%

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

   8. Taylor expanded in x around inf 18.7%

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

   9. Taylor expanded in t around 0 18.7%

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

       1. distribute-lft-out 18.7%

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

       2. sub-neg 18.7%

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

       3. metadata-eval 18.7%

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

       4. associate-/l* 18.7%

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

       5. +-commutative 18.7%

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

       6. +-commutative 18.7%

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

   11. Simplified 18.7%

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

   12. Taylor expanded in t around 0 56.3%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+209}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \end{array} \end{array} \]

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 (<= l 1.1e+210) (sqrt (/ (+ x -1.0) (+ x 1.0))) (* (sqrt x) (/ t_m l)))))

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 (l <= 1.1e+210) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = sqrt(x) * (t_m / l);
	}
	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 (l <= 1.1d+210) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = sqrt(x) * (t_m / l)
    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 (l <= 1.1e+210) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(x) * (t_m / l);
	}
	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 l <= 1.1e+210:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = math.sqrt(x) * (t_m / l)
	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 (l <= 1.1e+210)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	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 (l <= 1.1e+210)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = sqrt(x) * (t_m / l);
	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[l, 1.1e+210], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.09999999999999993e210

   1. Initial program 38.3%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in t around inf 40.6%

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

   6. Taylor expanded in t around 0 40.6%

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

   if 1.09999999999999993e210 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around 0 0.0%

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

      1. fma-define 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. associate-/l* 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. associate--l+ 18.7%

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

      8. sub-neg 18.7%

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

      9. metadata-eval 18.7%

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

      10. +-commutative 18.7%

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

      11. sub-neg 18.7%

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

      12. metadata-eval 18.7%

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

      13. +-commutative 18.7%

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

   7. Simplified 18.7%

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

   8. Taylor expanded in x around inf 18.7%

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

   9. Taylor expanded in t around 0 18.7%

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

       1. distribute-lft-out 18.7%

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

       2. sub-neg 18.7%

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

       3. metadata-eval 18.7%

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

       4. associate-/l* 18.7%

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

       5. +-commutative 18.7%

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

       6. +-commutative 18.7%

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

   11. Simplified 18.7%

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

   12. Taylor expanded in t around 0 56.3%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+210}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \end{array} \end{array} \]

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 (<= l 5.6e+210)
    (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))
    (* (sqrt x) (/ t_m l)))))

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 (l <= 5.6e+210) {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	} else {
		tmp = sqrt(x) * (t_m / l);
	}
	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 (l <= 5.6d+210) then
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    else
        tmp = sqrt(x) * (t_m / l)
    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 (l <= 5.6e+210) {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	} else {
		tmp = Math.sqrt(x) * (t_m / l);
	}
	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 l <= 5.6e+210:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	else:
		tmp = math.sqrt(x) * (t_m / l)
	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 (l <= 5.6e+210)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	else
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	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 (l <= 5.6e+210)
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	else
		tmp = sqrt(x) * (t_m / l);
	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[l, 5.6e+210], N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.6000000000000004e210

   1. Initial program 38.3%

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

   3. Simplified 32.7%

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

   5. Taylor expanded in t around inf 40.6%

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

   6. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   8. Simplified 40.3%

      \[\leadsto \color{blue}{1 - \frac{\frac{0.5}{-x} + 1}{x}} \]

   if 5.6000000000000004e210 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around 0 0.0%

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

      1. fma-define 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. associate-/l* 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. associate--l+ 18.7%

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

      8. sub-neg 18.7%

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

      9. metadata-eval 18.7%

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

      10. +-commutative 18.7%

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

      11. sub-neg 18.7%

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

      12. metadata-eval 18.7%

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

      13. +-commutative 18.7%

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

   7. Simplified 18.7%

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

   8. Taylor expanded in x around inf 18.7%

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

   9. Taylor expanded in t around 0 18.7%

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

       1. distribute-lft-out 18.7%

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

       2. sub-neg 18.7%

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

       3. metadata-eval 18.7%

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

       4. associate-/l* 18.7%

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

       5. +-commutative 18.7%

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

       6. +-commutative 18.7%

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

   11. Simplified 18.7%

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

   12. Taylor expanded in t around 0 56.3%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{+210}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1 + \frac{0.5}{x}}{x}\right) \end{array} \]

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 (/ (+ -1.0 (/ 0.5 x)) x))))

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 + ((-1.0 + (0.5 / x)) / x));
}

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 + (((-1.0d0) + (0.5d0 / x)) / x))
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 + ((-1.0 + (0.5 / x)) / x));
}

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 + ((-1.0 + (0.5 / x)) / x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x)))
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 + ((-1.0 + (0.5 / x)) / x));
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 * N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

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

3. Simplified 31.1%

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

5. Taylor expanded in t around inf 39.4%

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

6. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unsub-neg 0.0%

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

8. Simplified 39.2%

   \[\leadsto \color{blue}{1 - \frac{\frac{0.5}{-x} + 1}{x}} \]

9. Final simplification 39.2%

   \[\leadsto 1 + \frac{-1 + \frac{0.5}{x}}{x} \]
10. Add Preprocessing

Alternative 10: 76.3% accurate, 45.0× speedup

Math

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

FPCore

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 (/ -1.0 x))))

C

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

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 + ((-1.0d0) / x))
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 + (-1.0 / x));
}

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 + (-1.0 / x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
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 + (-1.0 / x));
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 * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

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

3. Simplified 31.1%

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

5. Taylor expanded in t around inf 39.4%

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

6. Taylor expanded in x around inf 39.2%

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

7. Final simplification 39.2%

   \[\leadsto 1 + \frac{-1}{x} \]
8. Add Preprocessing

Alternative 11: 75.7% accurate, 225.0× 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 36.3%

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

3. Simplified 31.1%

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

5. Taylor expanded in t around inf 39.4%

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

6. Taylor expanded in x around inf 39.1%

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

Reproduce

herbie shell --seed 2024110 
(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)))))