Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 98.8%

Time: 20.2s

Alternatives: 10

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: 33.5% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(\frac{\sqrt{2} \cdot \mathsf{hypot}\left(\ell, t\_2\right)}{\sqrt{x}}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= x 550.0)
      (sqrt (/ (+ x -1.0) (+ x 1.0)))
      (/ t_2 (hypot (/ (* (sqrt 2.0) (hypot l t_2)) (sqrt x)) 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 = sqrt(2.0) * t_m;
	double tmp;
	if (x <= 550.0) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_2 / hypot(((sqrt(2.0) * hypot(l, t_2)) / sqrt(x)), 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 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (x <= 550.0) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_2 / Math.hypot(((Math.sqrt(2.0) * Math.hypot(l, t_2)) / Math.sqrt(x)), 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 = math.sqrt(2.0) * t_m
	tmp = 0
	if x <= 550.0:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_2 / math.hypot(((math.sqrt(2.0) * math.hypot(l, t_2)) / math.sqrt(x)), 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(sqrt(2.0) * t_m)
	tmp = 0.0
	if (x <= 550.0)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_2 / hypot(Float64(Float64(sqrt(2.0) * hypot(l, t_2)) / sqrt(x)), 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 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (x <= 550.0)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_2 / hypot(((sqrt(2.0) * hypot(l, t_2)) / sqrt(x)), 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[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[x, 550.0], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$2 / N[Sqrt[N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 46.0%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t around inf 40.5%

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

   6. Taylor expanded in t around 0 40.6%

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

   if 550 < x

   1. Initial program 33.5%

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

      1. flip-+ 18.2%

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

      2. sub-neg 18.2%

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

      3. metadata-eval 18.2%

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

      4. div-inv 19.0%

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

      5. metadata-eval 19.0%

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

      6. fma-neg 19.0%

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

      7. metadata-eval 19.0%

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

   4. Applied egg-rr 19.0%

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

   5. Taylor expanded in x around inf 60.6%

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

      1. add-sqr-sqrt 60.6%

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

      2. add-sqr-sqrt 60.6%

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

      3. hypot-define 60.6%

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

   7. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

   9. Simplified 98.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\frac{\sqrt{2} \cdot \mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right)}{\sqrt{x}}, \sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\_m\right) \cdot {\left(2 \cdot \mathsf{fma}\left(t\_m, t\_m, \frac{\mathsf{fma}\left(2, {t\_m}^{2}, {\ell}^{2}\right)}{x}\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-168)
    (* (sqrt 2.0) (/ t_m (* (* (sqrt 2.0) l) (pow x -0.5))))
    (if (<= t_m 4e+77)
      (*
       (* (sqrt 2.0) t_m)
       (pow
        (* 2.0 (fma t_m t_m (/ (fma 2.0 (pow t_m 2.0) (pow l 2.0)) x)))
        -0.5))
      (sqrt (/ (+ x -1.0) (+ x 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 4.4e-168) {
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * pow(x, -0.5)));
	} else if (t_m <= 4e+77) {
		tmp = (sqrt(2.0) * t_m) * pow((2.0 * fma(t_m, t_m, (fma(2.0, pow(t_m, 2.0), pow(l, 2.0)) / x))), -0.5);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 4.4e-168)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(sqrt(2.0) * l) * (x ^ -0.5))));
	elseif (t_m <= 4e+77)
		tmp = Float64(Float64(sqrt(2.0) * t_m) * (Float64(2.0 * fma(t_m, t_m, Float64(fma(2.0, (t_m ^ 2.0), (l ^ 2.0)) / x))) ^ -0.5));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-168], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+77], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$m * t$95$m + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.3999999999999996e-168

   1. Initial program 31.4%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in x around inf 54.0%

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

   6. Taylor expanded in l around inf 18.3%

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

      1. associate-*l* 18.3%

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

   8. Simplified 18.3%

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

      1. *-un-lft-identity 18.3%

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

      2. inv-pow 18.3%

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

      3. sqrt-pow1 18.3%

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

      4. metadata-eval 18.3%

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

   10. Applied egg-rr 18.3%

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

       1. *-lft-identity 18.3%

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

       2. associate-*r* 18.3%

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

       3. *-commutative 18.3%

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

   12. Simplified 18.3%

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

   if 4.3999999999999996e-168 < t < 3.99999999999999993e77

   1. Initial program 69.5%

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

      1. flip-+ 29.1%

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

      2. sub-neg 29.1%

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

      3. metadata-eval 29.1%

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

      4. div-inv 29.3%

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

      5. metadata-eval 29.3%

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

      6. fma-neg 29.3%

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

      7. metadata-eval 29.3%

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

   4. Applied egg-rr 29.3%

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

   5. Taylor expanded in x around inf 92.8%

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

      1. div-inv 92.8%

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

      2. pow1/2 92.8%

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

      3. pow-flip 92.9%

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

      4. distribute-lft-out 92.9%

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

      5. fma-define 92.9%

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

      6. metadata-eval 92.9%

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

   7. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot t\right) \cdot {\left(2 \cdot \left(\frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x} + {t}^{2}\right)\right)}^{-0.5}} \]
   8. Step-by-step derivation

      1. +-commutative 92.9%

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

      2. unpow2 92.9%

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

      3. fma-define 92.9%

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

   9. Simplified 92.9%

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

   if 3.99999999999999993e77 < t

   1. Initial program 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 96.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\right) \cdot {\left(2 \cdot \mathsf{fma}\left(t, t, \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{t\_2 + 2 \cdot \frac{{\ell}^{2} + t\_2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.4e-168)
      (* (sqrt 2.0) (/ t_m (* (* (sqrt 2.0) l) (pow x -0.5))))
      (if (<= t_m 3e+77)
        (/ (* (sqrt 2.0) t_m) (sqrt (+ t_2 (* 2.0 (/ (+ (pow l 2.0) t_2) x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 4.4e-168) {
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * pow(x, -0.5)));
	} else if (t_m <= 3e+77) {
		tmp = (sqrt(2.0) * t_m) / sqrt((t_2 + (2.0 * ((pow(l, 2.0) + t_2) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 4.4d-168) then
        tmp = sqrt(2.0d0) * (t_m / ((sqrt(2.0d0) * l) * (x ** (-0.5d0))))
    else if (t_m <= 3d+77) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((t_2 + (2.0d0 * (((l ** 2.0d0) + t_2) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 4.4e-168) {
		tmp = Math.sqrt(2.0) * (t_m / ((Math.sqrt(2.0) * l) * Math.pow(x, -0.5)));
	} else if (t_m <= 3e+77) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((t_2 + (2.0 * ((Math.pow(l, 2.0) + t_2) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 4.4e-168:
		tmp = math.sqrt(2.0) * (t_m / ((math.sqrt(2.0) * l) * math.pow(x, -0.5)))
	elif t_m <= 3e+77:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((t_2 + (2.0 * ((math.pow(l, 2.0) + t_2) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.4e-168)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(sqrt(2.0) * l) * (x ^ -0.5))));
	elseif (t_m <= 3e+77)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(t_2 + Float64(2.0 * Float64(Float64((l ^ 2.0) + t_2) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 4.4e-168)
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * (x ^ -0.5)));
	elseif (t_m <= 3e+77)
		tmp = (sqrt(2.0) * t_m) / sqrt((t_2 + (2.0 * (((l ^ 2.0) + t_2) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-168], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+77], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.3999999999999996e-168

   1. Initial program 31.4%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in x around inf 54.0%

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

   6. Taylor expanded in l around inf 18.3%

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

      1. associate-*l* 18.3%

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

   8. Simplified 18.3%

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

      1. *-un-lft-identity 18.3%

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

      2. inv-pow 18.3%

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

      3. sqrt-pow1 18.3%

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

      4. metadata-eval 18.3%

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

   10. Applied egg-rr 18.3%

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

       1. *-lft-identity 18.3%

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

       2. associate-*r* 18.3%

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

       3. *-commutative 18.3%

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

   12. Simplified 18.3%

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

   if 4.3999999999999996e-168 < t < 2.9999999999999998e77

   1. Initial program 69.5%

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

      1. flip-+ 29.1%

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

      2. sub-neg 29.1%

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

      3. metadata-eval 29.1%

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

      4. div-inv 29.3%

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

      5. metadata-eval 29.3%

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

      6. fma-neg 29.3%

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

      7. metadata-eval 29.3%

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

   4. Applied egg-rr 29.3%

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

   5. Taylor expanded in x around inf 92.8%

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

   if 2.9999999999999998e77 < t

   1. Initial program 27.0%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in t around inf 95.8%

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

   6. Taylor expanded in t around 0 96.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{2 \cdot {t\_m}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-168)
    (* (sqrt 2.0) (/ t_m (* (* (sqrt 2.0) l) (pow x -0.5))))
    (if (<= t_m 3.4e-9)
      (/
       (* (sqrt 2.0) t_m)
       (sqrt (+ (* 2.0 (pow t_m 2.0)) (* 2.0 (/ (pow l 2.0) x)))))
      (sqrt (/ (+ x -1.0) (+ x 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.9e-168) {
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * pow(x, -0.5)));
	} else if (t_m <= 3.4e-9) {
		tmp = (sqrt(2.0) * t_m) / sqrt(((2.0 * pow(t_m, 2.0)) + (2.0 * (pow(l, 2.0) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.9d-168) then
        tmp = sqrt(2.0d0) * (t_m / ((sqrt(2.0d0) * l) * (x ** (-0.5d0))))
    else if (t_m <= 3.4d-9) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt(((2.0d0 * (t_m ** 2.0d0)) + (2.0d0 * ((l ** 2.0d0) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.9e-168) {
		tmp = Math.sqrt(2.0) * (t_m / ((Math.sqrt(2.0) * l) * Math.pow(x, -0.5)));
	} else if (t_m <= 3.4e-9) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt(((2.0 * Math.pow(t_m, 2.0)) + (2.0 * (Math.pow(l, 2.0) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.9e-168:
		tmp = math.sqrt(2.0) * (t_m / ((math.sqrt(2.0) * l) * math.pow(x, -0.5)))
	elif t_m <= 3.4e-9:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt(((2.0 * math.pow(t_m, 2.0)) + (2.0 * (math.pow(l, 2.0) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.9e-168)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(sqrt(2.0) * l) * (x ^ -0.5))));
	elseif (t_m <= 3.4e-9)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(2.0 * (t_m ^ 2.0)) + Float64(2.0 * Float64((l ^ 2.0) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.9e-168)
		tmp = sqrt(2.0) * (t_m / ((sqrt(2.0) * l) * (x ^ -0.5)));
	elseif (t_m <= 3.4e-9)
		tmp = (sqrt(2.0) * t_m) / sqrt(((2.0 * (t_m ^ 2.0)) + (2.0 * ((l ^ 2.0) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-168], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e-9], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.90000000000000012e-168

   1. Initial program 31.4%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in x around inf 54.0%

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

   6. Taylor expanded in l around inf 18.3%

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

      1. associate-*l* 18.3%

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

   8. Simplified 18.3%

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

      1. *-un-lft-identity 18.3%

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

      2. inv-pow 18.3%

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

      3. sqrt-pow1 18.3%

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

      4. metadata-eval 18.3%

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

   10. Applied egg-rr 18.3%

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

       1. *-lft-identity 18.3%

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

       2. associate-*r* 18.3%

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

       3. *-commutative 18.3%

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

   12. Simplified 18.3%

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

   if 3.90000000000000012e-168 < t < 3.3999999999999998e-9

   1. Initial program 62.0%

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

      1. flip-+ 25.0%

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

      2. sub-neg 25.0%

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

      3. metadata-eval 25.0%

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

      4. div-inv 24.9%

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

      5. metadata-eval 24.9%

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

      6. fma-neg 24.9%

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

      7. metadata-eval 24.9%

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

   4. Applied egg-rr 24.9%

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

   5. Taylor expanded in x around inf 88.9%

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

   6. Taylor expanded in t around 0 88.9%

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

   if 3.3999999999999998e-9 < t

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in t around inf 94.8%

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

   6. Taylor expanded in t around 0 95.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-168}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot {x}^{-0.5}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.82 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.82e-224)
    (* (sqrt 2.0) (/ t_m (* l (sqrt (/ 2.0 x)))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.82e-224) {
		tmp = sqrt(2.0) * (t_m / (l * sqrt((2.0 / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.82d-224) then
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt((2.0d0 / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.82e-224) {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt((2.0 / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.82e-224:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt((2.0 / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.82e-224)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.82e-224)
		tmp = sqrt(2.0) * (t_m / (l * sqrt((2.0 / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.82e-224], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8199999999999999e-224

   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 32.7%

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

   5. Taylor expanded in x around inf 55.8%

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

   6. Taylor expanded in l around inf 19.2%

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

      1. associate-*l* 19.1%

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

   8. Simplified 19.1%

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

      1. sqrt-unprod 19.1%

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

      2. pow1/2 19.1%

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

   10. Applied egg-rr 19.1%

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

       1. unpow1/2 19.1%

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

       2. associate-*r/ 19.1%

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

       3. metadata-eval 19.1%

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

   12. Simplified 19.1%

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

   if 1.8199999999999999e-224 < t

   1. Initial program 46.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around inf 87.3%

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

   6. Taylor expanded in t around 0 87.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.82 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.45e-224)
    (* (sqrt x) (/ t_m l))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.45e-224) {
		tmp = sqrt(x) * (t_m / l);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.45d-224) then
        tmp = sqrt(x) * (t_m / l)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.45e-224) {
		tmp = Math.sqrt(x) * (t_m / l);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.45e-224:
		tmp = math.sqrt(x) * (t_m / l)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.45e-224)
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.45e-224)
		tmp = sqrt(x) * (t_m / l);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-224], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.45e-224

   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 32.7%

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

   5. Taylor expanded in x around inf 55.8%

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

   6. Taylor expanded in l around inf 19.2%

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

      1. associate-*l* 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in t around 0 18.9%

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

   if 1.45e-224 < t

   1. Initial program 46.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around inf 87.3%

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

   6. Taylor expanded in t around 0 87.6%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.75e-223)
    (* (sqrt x) (/ t_m l))
    (+ 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) {
	double tmp;
	if (t_m <= 1.75e-223) {
		tmp = sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.75d-223) then
        tmp = sqrt(x) * (t_m / l)
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.75e-223) {
		tmp = Math.sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.75e-223:
		tmp = math.sqrt(x) * (t_m / l)
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.75e-223)
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.75e-223)
		tmp = sqrt(x) * (t_m / l);
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.75e-223], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.75000000000000005e-223

   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 32.7%

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

   5. Taylor expanded in x around inf 55.8%

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

   6. Taylor expanded in l around inf 19.2%

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

      1. associate-*l* 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in t around 0 18.9%

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

   if 1.75000000000000005e-223 < t

   1. Initial program 46.5%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around inf 87.3%

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

   6. Taylor expanded in t around 0 87.6%

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

      1. clear-num 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. sqrt-div 87.5%

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

      6. metadata-eval 87.5%

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

   8. Applied egg-rr 87.5%

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

   9. Taylor expanded in x around -inf 0.0%

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

   11. Simplified 86.9%

       \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% 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 38.7%

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

3. Simplified 31.9%

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

5. Taylor expanded in t around inf 38.9%

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

6. Taylor expanded in t around 0 39.0%

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

   1. clear-num 39.0%

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

   2. +-commutative 39.0%

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

   3. sub-neg 39.0%

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

   4. metadata-eval 39.0%

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

   5. sqrt-div 39.0%

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

   6. metadata-eval 39.0%

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

8. Applied egg-rr 39.0%

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

9. Taylor expanded in x around -inf 0.0%

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

11. Simplified 38.8%

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

12. Final simplification 38.8%

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

Alternative 9: 76.2% 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 38.7%

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

3. Simplified 31.9%

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

5. Taylor expanded in t around inf 38.9%

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

6. Taylor expanded in x around inf 38.7%

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

7. Final simplification 38.7%

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

Alternative 10: 75.5% 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 38.7%

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

3. Simplified 31.9%

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

5. Taylor expanded in t around inf 38.9%

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

6. Taylor expanded in x around inf 38.3%

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

Reproduce

herbie shell --seed 2024145 
(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)))))