Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 79.4%

Time: 25.6s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% 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: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+274}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e+274)
    (+ 1.0 (+ (/ 0.5 (pow x 2.0)) (- (/ -1.0 x) (/ 0.5 (pow x 3.0)))))
    (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (fma x 0.5 -0.5)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+274) {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-1.0 / x) - (0.5 / pow(x, 3.0))));
	} else {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e+274)
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-1.0 / x) - Float64(0.5 / (x ^ 3.0)))));
	else
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+274], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999998e274

   1. Initial program 43.8%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in l around 0 45.6%

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

      1. +-commutative 45.6%

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

      2. sub-neg 45.6%

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

      3. metadata-eval 45.6%

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

      4. +-commutative 45.6%

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

   7. Simplified 45.6%

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

   8. Taylor expanded in x around inf 45.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 45.7%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

      2. associate-*r/ 45.7%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      3. metadata-eval 45.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      4. +-commutative 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

      5. associate-*r/ 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right)\right) \]

      6. metadata-eval 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto \color{blue}{1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)} \]

   if 4.9999999999999998e274 < (*.f64 l l)

   1. Initial program 0.1%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in l around inf 4.4%

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

      1. *-commutative 4.4%

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

      2. associate--l+ 29.1%

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

      3. sub-neg 29.1%

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

      4. metadata-eval 29.1%

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

      5. +-commutative 29.1%

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

      6. sub-neg 29.1%

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

      7. metadata-eval 29.1%

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

      8. +-commutative 29.1%

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

   7. Simplified 29.1%

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

   8. Taylor expanded in x around 0 37.9%

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

   9. Taylor expanded in t around 0 37.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-/l* 37.9%

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

       2. *-commutative 37.9%

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

       3. fma-neg 37.9%

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

       4. metadata-eval 37.9%

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

       5. associate-*l* 40.7%

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

   11. Simplified 40.7%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+274}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{l\_m \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e+274)
    (+ 1.0 (+ (/ 0.5 (pow x 2.0)) (- (/ -1.0 x) (/ 0.5 (pow x 3.0)))))
    (/ (* t_m (sqrt 2.0)) (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+274) {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-1.0 / x) - (0.5 / pow(x, 3.0))));
	} else {
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 5d+274) then
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-1.0d0) / x) - (0.5d0 / (x ** 3.0d0))))
    else
        tmp = (t_m * sqrt(2.0d0)) / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0))))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+274) {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-1.0 / x) - (0.5 / Math.pow(x, 3.0))));
	} else {
		tmp = (t_m * Math.sqrt(2.0)) / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 5e+274:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-1.0 / x) - (0.5 / math.pow(x, 3.0))))
	else:
		tmp = (t_m * math.sqrt(2.0)) / (l_m * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0)))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e+274)
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-1.0 / x) - Float64(0.5 / (x ^ 3.0)))));
	else
		tmp = Float64(Float64(t_m * sqrt(2.0)) / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e+274)
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-1.0 / x) - (0.5 / (x ^ 3.0))));
	else
		tmp = (t_m * sqrt(2.0)) / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+274], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9999999999999998e274

   1. Initial program 43.8%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in l around 0 45.6%

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

      1. +-commutative 45.6%

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

      2. sub-neg 45.6%

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

      3. metadata-eval 45.6%

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

      4. +-commutative 45.6%

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

   7. Simplified 45.6%

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

   8. Taylor expanded in x around inf 45.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 45.7%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

      2. associate-*r/ 45.7%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      3. metadata-eval 45.7%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      4. +-commutative 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

      5. associate-*r/ 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right)\right) \]

      6. metadata-eval 45.7%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto \color{blue}{1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)} \]

   if 4.9999999999999998e274 < (*.f64 l l)

   1. Initial program 0.1%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in l around inf 4.9%

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

      1. associate--l+ 29.4%

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

      2. sub-neg 29.4%

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

      3. metadata-eval 29.4%

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

      4. +-commutative 29.4%

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

      5. sub-neg 29.4%

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

      6. metadata-eval 29.4%

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

      7. +-commutative 29.4%

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

   7. Simplified 29.4%

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

      1. associate-*r/ 29.4%

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

      2. +-commutative 29.4%

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

      3. sub-neg 29.4%

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

      4. +-commutative 29.4%

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

      5. metadata-eval 29.4%

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

   9. Applied egg-rr 29.4%

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

   10. Taylor expanded in x around inf 40.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+274}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 2.6e+138)
    (/ 1.0 (sqrt (/ (+ 1.0 x) (+ x -1.0))))
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.6e+138) {
		tmp = 1.0 / sqrt(((1.0 + x) / (x + -1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 2.6d+138) then
        tmp = 1.0d0 / sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 2.6e+138) {
		tmp = 1.0 / Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 2.6e+138:
		tmp = 1.0 / math.sqrt(((1.0 + x) / (x + -1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 2.6e+138)
		tmp = Float64(1.0 / sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 2.6e+138)
		tmp = 1.0 / sqrt(((1.0 + x) / (x + -1.0)));
	else
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 2.6e+138], N[(1.0 / N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.6000000000000001e138

   1. Initial program 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in l around 0 41.1%

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

      1. +-commutative 41.1%

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

      2. sub-neg 41.1%

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

      3. metadata-eval 41.1%

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

      4. +-commutative 41.1%

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

   7. Simplified 41.1%

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

      1. expm1-log1p-u 38.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      2. add-cube-cbrt 38.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      3. pow3 38.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      4. expm1-log1p-u 40.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{t \cdot \sqrt{2}}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      5. *-commutative 40.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{\sqrt{2} \cdot t}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   9. Applied egg-rr 40.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 40.3%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

       2. rem-cube-cbrt 41.1%

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

       3. +-commutative 41.1%

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

   11. Applied egg-rr 41.1%

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

       1. associate-/r* 41.1%

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

       2. *-inverses 41.1%

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

   13. Simplified 41.1%

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

   if 2.6000000000000001e138 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 9.8%

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

      1. associate--l+ 37.7%

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

      2. sub-neg 37.7%

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

      3. metadata-eval 37.7%

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

      4. +-commutative 37.7%

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

      5. sub-neg 37.7%

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

      6. metadata-eval 37.7%

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

      7. +-commutative 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in x around inf 64.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{l\_m \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.4e+138)
    (+ 1.0 (+ (/ 0.5 (pow x 2.0)) (- (/ -1.0 x) (/ 0.5 (pow x 3.0)))))
    (* (sqrt 2.0) (/ t_m (* l_m (sqrt (+ (/ 1.0 x) (/ 1.0 (+ x -1.0))))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.4e+138) {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-1.0 / x) - (0.5 / pow(x, 3.0))));
	} else {
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.4d+138) then
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-1.0d0) / x) - (0.5d0 / (x ** 3.0d0))))
    else
        tmp = sqrt(2.0d0) * (t_m / (l_m * sqrt(((1.0d0 / x) + (1.0d0 / (x + (-1.0d0)))))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.4e+138) {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-1.0 / x) - (0.5 / Math.pow(x, 3.0))));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l_m * Math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.4e+138:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-1.0 / x) - (0.5 / math.pow(x, 3.0))))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l_m * math.sqrt(((1.0 / x) + (1.0 / (x + -1.0))))))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.4e+138)
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-1.0 / x) - Float64(0.5 / (x ^ 3.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l_m * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0)))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.4e+138)
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-1.0 / x) - (0.5 / (x ^ 3.0))));
	else
		tmp = sqrt(2.0) * (t_m / (l_m * sqrt(((1.0 / x) + (1.0 / (x + -1.0))))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.4e+138], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e138

   1. Initial program 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in l around 0 41.1%

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

      1. +-commutative 41.1%

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

      2. sub-neg 41.1%

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

      3. metadata-eval 41.1%

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

      4. +-commutative 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in x around inf 41.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate--l+ 41.1%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

      2. associate-*r/ 41.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      3. metadata-eval 41.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) \]

      4. +-commutative 41.1%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

      5. associate-*r/ 41.1%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right)\right) \]

      6. metadata-eval 41.1%

         \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right)\right) \]

   10. Simplified 41.1%

       \[\leadsto \color{blue}{1 + \left(\frac{0.5}{{x}^{2}} - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right)} \]

   if 1.4e138 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 9.8%

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

      1. associate--l+ 37.7%

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

      2. sub-neg 37.7%

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

      3. metadata-eval 37.7%

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

      4. +-commutative 37.7%

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

      5. sub-neg 37.7%

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

      6. metadata-eval 37.7%

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

      7. +-commutative 37.7%

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

   7. Simplified 37.7%

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

   8. Taylor expanded in x around inf 64.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-1}{x} - \frac{0.5}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t\_m}{l\_m} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 9.5e+181)
    (/ 1.0 (sqrt (/ (+ 1.0 x) (+ x -1.0))))
    (* (sqrt 2.0) (* (/ t_m l_m) (sqrt (* 0.5 x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 9.5e+181) {
		tmp = 1.0 / sqrt(((1.0 + x) / (x + -1.0)));
	} else {
		tmp = sqrt(2.0) * ((t_m / l_m) * sqrt((0.5 * x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 9.5d+181) then
        tmp = 1.0d0 / sqrt(((1.0d0 + x) / (x + (-1.0d0))))
    else
        tmp = sqrt(2.0d0) * ((t_m / l_m) * sqrt((0.5d0 * x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 9.5e+181) {
		tmp = 1.0 / Math.sqrt(((1.0 + x) / (x + -1.0)));
	} else {
		tmp = Math.sqrt(2.0) * ((t_m / l_m) * Math.sqrt((0.5 * x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 9.5e+181:
		tmp = 1.0 / math.sqrt(((1.0 + x) / (x + -1.0)))
	else:
		tmp = math.sqrt(2.0) * ((t_m / l_m) * math.sqrt((0.5 * x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 9.5e+181)
		tmp = Float64(1.0 / sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(Float64(t_m / l_m) * sqrt(Float64(0.5 * x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 9.5e+181)
		tmp = 1.0 / sqrt(((1.0 + x) / (x + -1.0)));
	else
		tmp = sqrt(2.0) * ((t_m / l_m) * sqrt((0.5 * x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 9.5e+181], N[(1.0 / N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 9.50000000000000032e181

   1. Initial program 36.5%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in l around 0 41.4%

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

      1. +-commutative 41.4%

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

      2. sub-neg 41.4%

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

      3. metadata-eval 41.4%

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

      4. +-commutative 41.4%

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

   7. Simplified 41.4%

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

      1. expm1-log1p-u 38.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      3. pow3 38.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      4. expm1-log1p-u 40.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{t \cdot \sqrt{2}}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      5. *-commutative 40.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{\sqrt{2} \cdot t}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   9. Applied egg-rr 40.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 40.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

       2. rem-cube-cbrt 41.5%

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

       3. +-commutative 41.5%

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

   11. Applied egg-rr 41.5%

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

       1. associate-/r* 41.5%

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

       2. *-inverses 41.5%

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

   13. Simplified 41.5%

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

   if 9.50000000000000032e181 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 10.1%

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

      1. *-commutative 10.1%

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

      2. associate--l+ 40.8%

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

      3. sub-neg 40.8%

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

      4. metadata-eval 40.8%

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

      5. +-commutative 40.8%

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

      6. sub-neg 40.8%

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

      7. metadata-eval 40.8%

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

      8. +-commutative 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in x around inf 63.8%

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

      1. *-commutative 63.8%

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

   10. Simplified 63.8%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{t}{\ell} \cdot \sqrt{0.5 \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.45 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.45e+183)
    (/ 1.0 (sqrt (/ (+ 1.0 x) (+ x -1.0))))
    (* (/ t_m l_m) (sqrt (* 2.0 (fma x 0.5 -0.5)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.45e+183) {
		tmp = 1.0 / sqrt(((1.0 + x) / (x + -1.0)));
	} else {
		tmp = (t_m / l_m) * sqrt((2.0 * fma(x, 0.5, -0.5)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.45e+183)
		tmp = Float64(1.0 / sqrt(Float64(Float64(1.0 + x) / Float64(x + -1.0))));
	else
		tmp = Float64(Float64(t_m / l_m) * sqrt(Float64(2.0 * fma(x, 0.5, -0.5))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.45e+183], N[(1.0 / N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.45e183

   1. Initial program 36.5%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in l around 0 41.4%

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

      1. +-commutative 41.4%

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

      2. sub-neg 41.4%

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

      3. metadata-eval 41.4%

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

      4. +-commutative 41.4%

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

   7. Simplified 41.4%

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

      1. expm1-log1p-u 38.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      2. add-cube-cbrt 38.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      3. pow3 38.3%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      4. expm1-log1p-u 40.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{t \cdot \sqrt{2}}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      5. *-commutative 40.6%

         \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{\sqrt{2} \cdot t}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   9. Applied egg-rr 40.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 40.6%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

       2. rem-cube-cbrt 41.5%

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

       3. +-commutative 41.5%

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

   11. Applied egg-rr 41.5%

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

       1. associate-/r* 41.5%

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

       2. *-inverses 41.5%

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

   13. Simplified 41.5%

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

   if 1.45e183 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 10.1%

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

      1. *-commutative 10.1%

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

      2. associate--l+ 40.8%

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

      3. sub-neg 40.8%

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

      4. metadata-eval 40.8%

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

      5. +-commutative 40.8%

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

      6. sub-neg 40.8%

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

      7. metadata-eval 40.8%

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

      8. +-commutative 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in x around 0 63.8%

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

      1. pow1 63.8%

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

      2. associate-*r* 63.8%

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

      3. sqrt-unprod 63.9%

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

      4. *-commutative 63.9%

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

      5. fma-neg 63.9%

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

      6. metadata-eval 63.9%

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

   10. Applied egg-rr 63.9%

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

       1. unpow1 63.9%

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

       2. *-commutative 63.9%

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

   12. Simplified 63.9%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 + x}{x + -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 2.1× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5)))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * pow(((1.0 + x) / (x + -1.0)), -0.5);
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (((1.0d0 + x) / (x + (-1.0d0))) ** (-0.5d0))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * Math.pow(((1.0 + x) / (x + -1.0)), -0.5);
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * math.pow(((1.0 + x) / (x + -1.0)), -0.5)

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * (Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (((1.0 + x) / (x + -1.0)) ^ -0.5);
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 33.0%

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

5. Taylor expanded in l around 0 38.9%

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

   1. +-commutative 38.9%

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

   2. sub-neg 38.9%

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

   3. metadata-eval 38.9%

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

   4. +-commutative 38.9%

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

7. Simplified 38.9%

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

   1. expm1-log1p-u 36.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   2. add-cube-cbrt 36.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)} \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   3. pow3 36.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \sqrt{2}\right)\right)}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   4. expm1-log1p-u 38.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{t \cdot \sqrt{2}}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

   5. *-commutative 38.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{{\left(\sqrt[3]{\color{blue}{\sqrt{2} \cdot t}}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

9. Applied egg-rr 38.2%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3}} \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]
10. Step-by-step derivation

    1. associate-*r/ 38.2%

       \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{{\left(\sqrt[3]{\sqrt{2} \cdot t}\right)}^{3} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

    2. rem-cube-cbrt 39.0%

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

    3. +-commutative 39.0%

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

11. Applied egg-rr 39.0%

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

    1. associate-/r* 39.0%

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

    2. *-inverses 39.0%

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

13. Simplified 39.0%

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

    1. *-un-lft-identity 39.0%

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

    2. pow1/2 39.0%

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

    3. pow-flip 39.0%

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

    4. +-commutative 39.0%

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

    5. metadata-eval 39.0%

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

15. Applied egg-rr 39.0%

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

    1. *-lft-identity 39.0%

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

    2. +-commutative 39.0%

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

    3. +-commutative 39.0%

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

17. Simplified 39.0%

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

18. Final simplification 39.0%

    \[\leadsto {\left(\frac{1 + x}{x + -1}\right)}^{-0.5} \]
19. Add Preprocessing

Alternative 8: 76.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $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 33.0%

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

5. Taylor expanded in l around 0 38.9%

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

   1. +-commutative 38.9%

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

   2. sub-neg 38.9%

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

   3. metadata-eval 38.9%

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

   4. +-commutative 38.9%

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

7. Simplified 38.9%

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

8. Taylor expanded in t around 0 39.0%

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

9. Final simplification 39.0%

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

Alternative 9: 75.7% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(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 33.0%

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

5. Taylor expanded in l around 0 38.9%

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

   1. +-commutative 38.9%

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

   2. sub-neg 38.9%

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

   3. metadata-eval 38.9%

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

   4. +-commutative 38.9%

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

7. Simplified 38.9%

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

8. Taylor expanded in x around inf 38.9%

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

9. Final simplification 38.9%

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

Alternative 10: 75.0% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in l around 0 38.9%

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

   1. +-commutative 38.9%

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

   2. sub-neg 38.9%

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

   3. metadata-eval 38.9%

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

   4. +-commutative 38.9%

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

7. Simplified 38.9%

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

8. Taylor expanded in x around inf 38.7%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 38.7%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))))