Toniolo and Linder, Equation (7)

Percentage Accurate: 33.1% → 98.8%

Time: 22.1s

Alternatives: 13

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: 33.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 30000:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\mathsf{hypot}\left(\ell, t\_2\right)}{\sqrt{x}}, t\_2\right)}\\ \end{array} \end{array} \end{array} \]

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 30000.0)
      (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 <= 30000.0) {
		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 <= 30000.0) {
		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 <= 30000.0:
		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 <= 30000.0)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_2 / hypot(Float64(sqrt(2.0) * Float64(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 <= 30000.0)
		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, 30000.0], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$2 / N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e4

   1. Initial program 40.0%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in t around inf 46.8%

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

      1. associate-*l* 46.8%

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

      2. sub-neg 46.8%

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

      3. metadata-eval 46.8%

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

      4. +-commutative 46.8%

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

      5. +-commutative 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in t around 0 46.9%

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

   if 3e4 < x

   1. Initial program 28.5%

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

      1. flip-+ 14.8%

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

      2. sub-neg 14.8%

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

      3. metadata-eval 14.8%

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

      4. div-inv 14.8%

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

      5. metadata-eval 14.8%

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

      6. fma-neg 14.8%

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

      7. metadata-eval 14.8%

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

   4. Applied egg-rr 14.8%

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

   5. Taylor expanded in x around inf 58.2%

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

      1. add-sqr-sqrt 58.2%

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

      2. add-sqr-sqrt 58.2%

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

      3. hypot-define 58.2%

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30000:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right)}{\sqrt{x}}, \sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_3}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 3.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t\_3\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_3}{\sqrt{t\_2 + 2 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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_3 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 9.5e-237)
      (/ t_3 (* l (* (sqrt 2.0) (sqrt (/ 1.0 x)))))
      (if (<= t_m 3.8e-203)
        (/
         1.0
         (/ (hypot (* (hypot l t_3) (sqrt (/ (+ x 1.0) (+ x -1.0)))) l) t_3))
        (if (<= t_m 1.6e+48)
          (/ t_3 (sqrt (+ t_2 (* 2.0 (/ (+ t_2 (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 t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = t_3 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	} else if (t_m <= 3.8e-203) {
		tmp = 1.0 / (hypot((hypot(l, t_3) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_3);
	} else if (t_m <= 1.6e+48) {
		tmp = t_3 / sqrt((t_2 + (2.0 * ((t_2 + pow(l, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	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 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = t_3 / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x))));
	} else if (t_m <= 3.8e-203) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l, t_3) * Math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_3);
	} else if (t_m <= 1.6e+48) {
		tmp = t_3 / Math.sqrt((t_2 + (2.0 * ((t_2 + 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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 9.5e-237:
		tmp = t_3 / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x))))
	elif t_m <= 3.8e-203:
		tmp = 1.0 / (math.hypot((math.hypot(l, t_3) * math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_3)
	elif t_m <= 1.6e+48:
		tmp = t_3 / math.sqrt((t_2 + (2.0 * ((t_2 + 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)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 9.5e-237)
		tmp = Float64(t_3 / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x)))));
	elseif (t_m <= 3.8e-203)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l, t_3) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))), l) / t_3));
	elseif (t_m <= 1.6e+48)
		tmp = Float64(t_3 / sqrt(Float64(t_2 + Float64(2.0 * Float64(Float64(t_2 + (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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 9.5e-237)
		tmp = t_3 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	elseif (t_m <= 3.8e-203)
		tmp = 1.0 / (hypot((hypot(l, t_3) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_3);
	elseif (t_m <= 1.6e+48)
		tmp = t_3 / sqrt((t_2 + (2.0 * ((t_2 + (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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-237], N[(t$95$3 / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.8e-203], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l ^ 2 + t$95$3 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+48], N[(t$95$3 / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $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 4 regimes

2. if t < 9.4999999999999998e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 28.3%

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

      1. associate-*r/ 28.3%

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

      2. metadata-eval 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   10. Simplified 18.1%

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

   if 9.4999999999999998e-237 < t < 3.80000000000000025e-203

   1. Initial program 3.0%

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 84.4%

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

   if 3.80000000000000025e-203 < t < 1.6000000000000001e48

   1. Initial program 45.6%

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

      1. flip-+ 27.2%

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

      2. sub-neg 27.2%

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

      3. metadata-eval 27.2%

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

      4. div-inv 27.3%

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

      5. metadata-eval 27.3%

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

      6. fma-neg 27.3%

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

      7. metadata-eval 27.3%

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

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in x around inf 75.8%

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

   if 1.6000000000000001e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in t around inf 98.1%

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

      1. associate-*l* 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{\sqrt{2} \cdot t}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t\_2\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t\_2}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot {t\_m}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 8.8e-237)
      (/ t_2 (* l (* (sqrt 2.0) (sqrt (/ 1.0 x)))))
      (if (<= t_m 7.5e-205)
        (/
         1.0
         (/ (hypot (* (hypot l t_2) (sqrt (/ (+ x 1.0) (+ x -1.0)))) l) t_2))
        (if (<= t_m 2e+48)
          (/ t_2 (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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 8.8e-237) {
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	} else if (t_m <= 7.5e-205) {
		tmp = 1.0 / (hypot((hypot(l, t_2) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_2);
	} else if (t_m <= 2e+48) {
		tmp = t_2 / 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;
}

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 (t_m <= 8.8e-237) {
		tmp = t_2 / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x))));
	} else if (t_m <= 7.5e-205) {
		tmp = 1.0 / (Math.hypot((Math.hypot(l, t_2) * Math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_2);
	} else if (t_m <= 2e+48) {
		tmp = t_2 / 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):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 8.8e-237:
		tmp = t_2 / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x))))
	elif t_m <= 7.5e-205:
		tmp = 1.0 / (math.hypot((math.hypot(l, t_2) * math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_2)
	elif t_m <= 2e+48:
		tmp = t_2 / 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 8.8e-237)
		tmp = Float64(t_2 / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x)))));
	elseif (t_m <= 7.5e-205)
		tmp = Float64(1.0 / Float64(hypot(Float64(hypot(l, t_2) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))), l) / t_2));
	elseif (t_m <= 2e+48)
		tmp = Float64(t_2 / 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)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 8.8e-237)
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	elseif (t_m <= 7.5e-205)
		tmp = 1.0 / (hypot((hypot(l, t_2) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_2);
	elseif (t_m <= 2e+48)
		tmp = t_2 / 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.8e-237], N[(t$95$2 / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-205], N[(1.0 / N[(N[Sqrt[N[(N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+48], N[(t$95$2 / 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 4 regimes

2. if t < 8.79999999999999992e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 28.3%

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

      1. associate-*r/ 28.3%

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

      2. metadata-eval 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   10. Simplified 18.1%

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

   if 8.79999999999999992e-237 < t < 7.4999999999999996e-205

   1. Initial program 3.0%

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 84.4%

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

   if 7.4999999999999996e-205 < t < 2.00000000000000009e48

   1. Initial program 45.6%

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

      1. flip-+ 27.2%

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

      2. sub-neg 27.2%

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

      3. metadata-eval 27.2%

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

      4. div-inv 27.3%

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

      5. metadata-eval 27.3%

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

      6. fma-neg 27.3%

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

      7. metadata-eval 27.3%

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

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in x around inf 75.8%

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

   6. Taylor expanded in t around 0 74.9%

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

   if 2.00000000000000009e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in t around inf 98.1%

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

      1. associate-*l* 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{\sqrt{2} \cdot t}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, t\_2\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t\_m}}\\ \mathbf{elif}\;t\_m \leq 1.36 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot {t\_m}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.25e-237)
      (/ t_2 (* l (* (sqrt 2.0) (sqrt (/ 1.0 x)))))
      (if (<= t_m 8e-205)
        (/
         (sqrt 2.0)
         (/ (hypot (* (hypot l t_2) (sqrt (/ (+ x 1.0) (+ x -1.0)))) l) t_m))
        (if (<= t_m 1.36e+48)
          (/ t_2 (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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.25e-237) {
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	} else if (t_m <= 8e-205) {
		tmp = sqrt(2.0) / (hypot((hypot(l, t_2) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_m);
	} else if (t_m <= 1.36e+48) {
		tmp = t_2 / 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;
}

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 (t_m <= 2.25e-237) {
		tmp = t_2 / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x))));
	} else if (t_m <= 8e-205) {
		tmp = Math.sqrt(2.0) / (Math.hypot((Math.hypot(l, t_2) * Math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_m);
	} else if (t_m <= 1.36e+48) {
		tmp = t_2 / 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):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 2.25e-237:
		tmp = t_2 / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x))))
	elif t_m <= 8e-205:
		tmp = math.sqrt(2.0) / (math.hypot((math.hypot(l, t_2) * math.sqrt(((x + 1.0) / (x + -1.0)))), l) / t_m)
	elif t_m <= 1.36e+48:
		tmp = t_2 / 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.25e-237)
		tmp = Float64(t_2 / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x)))));
	elseif (t_m <= 8e-205)
		tmp = Float64(sqrt(2.0) / Float64(hypot(Float64(hypot(l, t_2) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))), l) / t_m));
	elseif (t_m <= 1.36e+48)
		tmp = Float64(t_2 / 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)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 2.25e-237)
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	elseif (t_m <= 8e-205)
		tmp = sqrt(2.0) / (hypot((hypot(l, t_2) * sqrt(((x + 1.0) / (x + -1.0)))), l) / t_m);
	elseif (t_m <= 1.36e+48)
		tmp = t_2 / 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.25e-237], N[(t$95$2 / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e-205], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.36e+48], N[(t$95$2 / 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 4 regimes

2. if t < 2.25000000000000005e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 28.3%

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

      1. associate-*r/ 28.3%

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

      2. metadata-eval 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   10. Simplified 18.1%

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

   if 2.25000000000000005e-237 < t < 8e-205

   1. Initial program 3.0%

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 84.4%

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

   if 8e-205 < t < 1.3599999999999999e48

   1. Initial program 45.6%

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

      1. flip-+ 27.2%

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

      2. sub-neg 27.2%

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

      3. metadata-eval 27.2%

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

      4. div-inv 27.3%

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

      5. metadata-eval 27.3%

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

      6. fma-neg 27.3%

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

      7. metadata-eval 27.3%

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

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in x around inf 75.8%

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

   6. Taylor expanded in t around 0 74.9%

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

   if 1.3599999999999999e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in t around inf 98.1%

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

      1. associate-*l* 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}{t}}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{t\_2}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{-202}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_2}{\sqrt{2 \cdot {t\_m}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 9.5e-237)
      (/ t_2 (* l (* (sqrt 2.0) (sqrt (/ 1.0 x)))))
      (if (<= t_m 6.5e-202)
        (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))
        (if (<= t_m 1.4e+48)
          (/ t_2 (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 t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	} else if (t_m <= 6.5e-202) {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	} else if (t_m <= 1.4e+48) {
		tmp = t_2 / 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) :: t_2
    real(8) :: tmp
    t_2 = sqrt(2.0d0) * t_m
    if (t_m <= 9.5d-237) then
        tmp = t_2 / (l * (sqrt(2.0d0) * sqrt((1.0d0 / x))))
    else if (t_m <= 6.5d-202) then
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    else if (t_m <= 1.4d+48) then
        tmp = t_2 / 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 t_2 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 9.5e-237) {
		tmp = t_2 / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x))));
	} else if (t_m <= 6.5e-202) {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	} else if (t_m <= 1.4e+48) {
		tmp = t_2 / 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):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 9.5e-237:
		tmp = t_2 / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x))))
	elif t_m <= 6.5e-202:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	elif t_m <= 1.4e+48:
		tmp = t_2 / 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)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 9.5e-237)
		tmp = Float64(t_2 / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x)))));
	elseif (t_m <= 6.5e-202)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	elseif (t_m <= 1.4e+48)
		tmp = Float64(t_2 / 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)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 9.5e-237)
		tmp = t_2 / (l * (sqrt(2.0) * sqrt((1.0 / x))));
	elseif (t_m <= 6.5e-202)
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	elseif (t_m <= 1.4e+48)
		tmp = t_2 / 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_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-237], N[(t$95$2 / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e-202], N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+48], N[(t$95$2 / 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 4 regimes

2. if t < 9.4999999999999998e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 28.3%

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

      1. associate-*r/ 28.3%

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

      2. metadata-eval 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   10. Simplified 18.1%

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

   if 9.4999999999999998e-237 < t < 6.49999999999999956e-202

   1. Initial program 3.0%

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

   3. Simplified 2.6%

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

   5. Taylor expanded in t around inf 84.2%

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

      1. associate-*l* 84.2%

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

      2. sub-neg 84.2%

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

      3. metadata-eval 84.2%

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

      4. +-commutative 84.2%

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

      5. +-commutative 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   9. 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%

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

   10. Simplified 84.2%

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

   if 6.49999999999999956e-202 < t < 1.40000000000000006e48

   1. Initial program 45.6%

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

      1. flip-+ 27.2%

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

      2. sub-neg 27.2%

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

      3. metadata-eval 27.2%

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

      4. div-inv 27.3%

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

      5. metadata-eval 27.3%

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

      6. fma-neg 27.3%

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

      7. metadata-eval 27.3%

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

   4. Applied egg-rr 27.3%

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

   5. Taylor expanded in x around inf 75.8%

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

   6. Taylor expanded in t around 0 74.9%

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

   if 1.40000000000000006e48 < t

   1. Initial program 34.5%

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

   3. Simplified 34.6%

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

   5. Taylor expanded in t around inf 98.1%

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

      1. associate-*l* 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in t around 0 98.3%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-202}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 3.3e-237)
    (/ (* (sqrt 2.0) t_m) (* l (* (sqrt 2.0) (sqrt (/ 1.0 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 <= 3.3e-237) {
		tmp = (sqrt(2.0) * t_m) / (l * (sqrt(2.0) * sqrt((1.0 / 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 <= 3.3d-237) then
        tmp = (sqrt(2.0d0) * t_m) / (l * (sqrt(2.0d0) * sqrt((1.0d0 / 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 <= 3.3e-237) {
		tmp = (Math.sqrt(2.0) * t_m) / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / 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 <= 3.3e-237:
		tmp = (math.sqrt(2.0) * t_m) / (l * (math.sqrt(2.0) * math.sqrt((1.0 / 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 <= 3.3e-237)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / 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 <= 3.3e-237)
		tmp = (sqrt(2.0) * t_m) / (l * (sqrt(2.0) * sqrt((1.0 / 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, 3.3e-237], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / 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 < 3.3000000000000001e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 28.3%

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

      1. associate-*r/ 28.3%

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

      2. metadata-eval 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in l around inf 18.1%

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

      1. associate-*l* 18.1%

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

   10. Simplified 18.1%

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

   if 3.3000000000000001e-237 < t

   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 32.5%

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

   5. Taylor expanded in t around inf 77.1%

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

      1. associate-*l* 77.1%

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

      2. sub-neg 77.1%

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

      3. metadata-eval 77.1%

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

      4. +-commutative 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 2.25e-237)
    (/ (* (sqrt 2.0) t_m) (sqrt (* 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 <= 2.25e-237) {
		tmp = (sqrt(2.0) * t_m) / sqrt((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 <= 2.25d-237) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((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 <= 2.25e-237) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((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 <= 2.25e-237:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((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 <= 2.25e-237)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(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 <= 2.25e-237)
		tmp = (sqrt(2.0) * t_m) / sqrt((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, 2.25e-237], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $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 < 2.25000000000000005e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 52.9%

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

   6. Taylor expanded in t around 0 20.3%

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

   if 2.25000000000000005e-237 < t

   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 32.5%

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

   5. Taylor expanded in t around inf 77.1%

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

      1. associate-*l* 77.1%

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

      2. sub-neg 77.1%

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

      3. metadata-eval 77.1%

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

      4. +-commutative 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{x} \cdot \left(t\_m \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 4.6e-237)
    (* (sqrt x) (* t_m (* (sqrt 0.5) (/ (sqrt 2.0) 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 <= 4.6e-237) {
		tmp = sqrt(x) * (t_m * (sqrt(0.5) * (sqrt(2.0) / 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 <= 4.6d-237) then
        tmp = sqrt(x) * (t_m * (sqrt(0.5d0) * (sqrt(2.0d0) / 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 <= 4.6e-237) {
		tmp = Math.sqrt(x) * (t_m * (Math.sqrt(0.5) * (Math.sqrt(2.0) / 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 <= 4.6e-237:
		tmp = math.sqrt(x) * (t_m * (math.sqrt(0.5) * (math.sqrt(2.0) / 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 <= 4.6e-237)
		tmp = Float64(sqrt(x) * Float64(t_m * Float64(sqrt(0.5) * Float64(sqrt(2.0) / 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 <= 4.6e-237)
		tmp = sqrt(x) * (t_m * (sqrt(0.5) * (sqrt(2.0) / 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, 4.6e-237], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / l), $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 < 4.60000000000000023e-237

   1. Initial program 28.6%

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

      1. flip-+ 15.5%

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

      2. sub-neg 15.5%

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

      3. metadata-eval 15.5%

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

      4. div-inv 15.5%

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

      5. metadata-eval 15.5%

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

      6. fma-neg 15.5%

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

      7. metadata-eval 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in x around inf 52.9%

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

      1. add-sqr-sqrt 52.7%

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

      2. pow2 52.7%

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

      3. pow1/2 52.7%

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

      4. sqrt-pow1 52.7%

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

      5. metadata-eval 52.7%

         \[\leadsto \frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2} \cdot t}{\sqrt{2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + 2 \cdot {t}^{2}}} \]

   7. Applied egg-rr 52.7%

      \[\leadsto \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}} \cdot t}{\sqrt{2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + 2 \cdot {t}^{2}}} \]

   8. Taylor expanded in t around 0 14.8%

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

      1. associate-/l* 14.8%

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

      2. associate-/l* 14.8%

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

   10. Simplified 14.8%

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

   if 4.60000000000000023e-237 < t

   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 32.5%

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

   5. Taylor expanded in t around inf 77.1%

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

      1. associate-*l* 77.1%

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

      2. sub-neg 77.1%

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

      3. metadata-eval 77.1%

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

      4. +-commutative 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{x} \cdot \left(t \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{1}{x + -1} + \left(-1 + \frac{x}{x + -1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

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 2.5e-237)
    (*
     (sqrt 2.0)
     (/ t_m (* l (sqrt (+ (/ 1.0 (+ x -1.0)) (+ -1.0 (/ x (+ x -1.0))))))))
    (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 <= 2.5e-237) {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + (-1.0 + (x / (x + -1.0)))))));
	} 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 <= 2.5d-237) then
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((1.0d0 / (x + (-1.0d0))) + ((-1.0d0) + (x / (x + (-1.0d0))))))))
    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 <= 2.5e-237) {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((1.0 / (x + -1.0)) + (-1.0 + (x / (x + -1.0)))))));
	} 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 <= 2.5e-237:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((1.0 / (x + -1.0)) + (-1.0 + (x / (x + -1.0)))))))
	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 <= 2.5e-237)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 + Float64(x / Float64(x + -1.0))))))));
	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 <= 2.5e-237)
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + (-1.0 + (x / (x + -1.0)))))));
	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, 2.5e-237], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 2.5000000000000001e-237

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

      1. add-exp-log 16.2%

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

      2. +-commutative 16.2%

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

      3. log1p-define 16.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{e^{\color{blue}{\mathsf{log1p}\left(x\right)}}}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   6. Applied egg-rr 16.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{e^{\mathsf{log1p}\left(x\right)}}}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   7. Taylor expanded in l around inf 3.4%

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

      1. associate--l+ 10.9%

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

      2. sub-neg 10.9%

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

      3. metadata-eval 10.9%

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

      4. sub-neg 10.9%

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

      5. metadata-eval 10.9%

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

   9. Simplified 10.9%

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

   if 2.5000000000000001e-237 < t

   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 32.5%

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

   5. Taylor expanded in t around inf 77.1%

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

      1. associate-*l* 77.1%

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

      2. sub-neg 77.1%

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

      3. metadata-eval 77.1%

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

      4. +-commutative 77.1%

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

      5. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t around 0 77.2%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x + -1} + \left(-1 + \frac{x}{x + -1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 2.1× speedup

Math

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

FPCore

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 (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) {
	return t_s * sqrt(((x + -1.0) / (x + 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 * sqrt(((x + (-1.0d0)) / (x + 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 * Math.sqrt(((x + -1.0) / (x + 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 * math.sqrt(((x + -1.0) / (x + 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 * sqrt(Float64(Float64(x + -1.0) / Float64(x + 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 * sqrt(((x + -1.0) / (x + 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 * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

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 36.4%

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

   1. associate-*l* 36.4%

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

   2. sub-neg 36.4%

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

   3. metadata-eval 36.4%

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

   4. +-commutative 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in t around 0 36.4%

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

9. Final simplification 36.4%

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

Alternative 11: 76.3% 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 33.0%

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

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 36.4%

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

   1. associate-*l* 36.4%

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

   2. sub-neg 36.4%

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

   3. metadata-eval 36.4%

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

   4. +-commutative 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{0.5 \cdot \frac{2 + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x \cdot {\left(\sqrt{-1}\right)}^{2}} - \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
9. 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%

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

10. Simplified 36.4%

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

11. Final simplification 36.4%

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

Alternative 12: 76.1% 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 33.0%

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

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 36.4%

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

   1. associate-*l* 36.4%

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

   2. sub-neg 36.4%

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

   3. metadata-eval 36.4%

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

   4. +-commutative 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around inf 36.4%

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

9. Final simplification 36.4%

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

Alternative 13: 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 33.0%

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

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 36.4%

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

   1. associate-*l* 36.4%

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

   2. sub-neg 36.4%

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

   3. metadata-eval 36.4%

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

   4. +-commutative 36.4%

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

   5. +-commutative 36.4%

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

7. Simplified 36.4%

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

8. Taylor expanded in x around inf 36.1%

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

Reproduce

herbie shell --seed 2024132 
(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)))))