Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 87.3%

Time: 32.8s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+150}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -8.3 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-164}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 96000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* t (sqrt 2.0)) (sqrt (* 2.0 (+ (* l (/ l x)) (* t t))))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.7e+150)
     (- t_2)
     (if (<= t -8.3e-203)
       t_1
       (if (<= t 6.9e-164)
         (/ (* t (sqrt x)) l)
         (if (<= t 96000000.0) t_1 t_2))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (t * sqrt(2.0)) / sqrt((2.0 * ((l * (l / x)) + (t * t))));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.7e+150) {
		tmp = -t_2;
	} else if (t <= -8.3e-203) {
		tmp = t_1;
	} else if (t <= 6.9e-164) {
		tmp = (t * sqrt(x)) / l;
	} else if (t <= 96000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t * sqrt(2.0d0)) / sqrt((2.0d0 * ((l * (l / x)) + (t * t))))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-1.7d+150)) then
        tmp = -t_2
    else if (t <= (-8.3d-203)) then
        tmp = t_1
    else if (t <= 6.9d-164) then
        tmp = (t * sqrt(x)) / l
    else if (t <= 96000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (t * Math.sqrt(2.0)) / Math.sqrt((2.0 * ((l * (l / x)) + (t * t))));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.7e+150) {
		tmp = -t_2;
	} else if (t <= -8.3e-203) {
		tmp = t_1;
	} else if (t <= 6.9e-164) {
		tmp = (t * Math.sqrt(x)) / l;
	} else if (t <= 96000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (t * math.sqrt(2.0)) / math.sqrt((2.0 * ((l * (l / x)) + (t * t))))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1.7e+150:
		tmp = -t_2
	elif t <= -8.3e-203:
		tmp = t_1
	elif t <= 6.9e-164:
		tmp = (t * math.sqrt(x)) / l
	elif t <= 96000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(t * sqrt(2.0)) / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(t * t)))))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.7e+150)
		tmp = Float64(-t_2);
	elseif (t <= -8.3e-203)
		tmp = t_1;
	elseif (t <= 6.9e-164)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	elseif (t <= 96000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (t * sqrt(2.0)) / sqrt((2.0 * ((l * (l / x)) + (t * t))));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1.7e+150)
		tmp = -t_2;
	elseif (t <= -8.3e-203)
		tmp = t_1;
	elseif (t <= 6.9e-164)
		tmp = (t * sqrt(x)) / l;
	elseif (t <= 96000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.7e+150], (-t$95$2), If[LessEqual[t, -8.3e-203], t$95$1, If[LessEqual[t, 6.9e-164], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t, 96000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.69999999999999991e150

   1. Initial program 2.3%

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

      1. associate-*l/ 2.3%

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

      2. *-commutative 2.3%

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

      3. fma-neg 2.3%

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

      4. sqr-neg 2.3%

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

      5. fma-neg 2.3%

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

   3. Simplified 2.3%

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

   5. Applied egg-rr 88.0%

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

   6. Taylor expanded in t around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

      5. +-commutative 97.6%

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

   8. Simplified 97.6%

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

   if -1.69999999999999991e150 < t < -8.29999999999999985e-203 or 6.89999999999999995e-164 < t < 9.6e7

   1. Initial program 50.3%

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

      1. fma-neg 50.3%

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

      2. sqr-neg 50.3%

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

      3. fma-neg 50.3%

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

      4. sqr-neg 50.3%

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

      5. sqr-neg 50.3%

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

      6. sqr-neg 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in x around inf 77.4%

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

      1. distribute-lft-out 77.4%

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

      2. fma-def 77.4%

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

      3. unpow2 77.4%

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

      4. unpow2 77.4%

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

      5. unpow2 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. unpow2 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. associate-/l* 88.3%

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

   11. Applied egg-rr 88.3%

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

       1. *-lft-identity 88.3%

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

       2. associate-/r/ 88.4%

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

   13. Simplified 88.4%

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

   if -8.29999999999999985e-203 < t < 6.89999999999999995e-164

   1. Initial program 7.2%

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

      1. fma-neg 7.3%

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

      2. sqr-neg 7.3%

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

      3. fma-neg 7.2%

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

      4. sqr-neg 7.2%

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

      5. sqr-neg 7.2%

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

      6. sqr-neg 7.2%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in x around inf 52.8%

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

      1. distribute-lft-out 52.8%

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

      2. fma-def 52.8%

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

      3. unpow2 52.8%

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

      4. unpow2 52.8%

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

      5. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in t around 0 37.7%

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

      1. associate-*l/ 41.8%

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

   9. Applied egg-rr 41.8%

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

   if 9.6e7 < t

   1. Initial program 40.6%

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

      1. associate-*l/ 40.6%

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

      2. *-commutative 40.6%

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

      3. fma-neg 40.6%

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

      4. sqr-neg 40.6%

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

      5. fma-neg 40.6%

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

   3. Simplified 40.6%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in l around 0 97.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+150}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -8.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-164}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 96000000:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.8e-139)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 8.5e-64) (/ (* t (sqrt x)) l) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 8.5e-64) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.8d-139)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 8.5d-64) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.8e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 8.5e-64) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.8e-139:
		tmp = -1.0 + (1.0 / x)
	elif t <= 8.5e-64:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.8e-139)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 8.5e-64)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.8e-139)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 8.5e-64)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.8e-139], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-64], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.80000000000000002e-139

   1. Initial program 37.4%

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

      1. associate-*l/ 37.4%

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

      2. *-commutative 37.4%

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

      3. fma-neg 37.4%

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

      4. sqr-neg 37.4%

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

      5. fma-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Applied egg-rr 71.5%

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

   6. Taylor expanded in l around 0 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 86.2%

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

   9. Simplified 86.2%

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

   if -1.80000000000000002e-139 < t < 8.49999999999999996e-64

   1. Initial program 11.7%

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

      1. fma-neg 11.8%

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

      2. sqr-neg 11.8%

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

      3. fma-neg 11.7%

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

      4. sqr-neg 11.7%

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

      5. sqr-neg 11.7%

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

      6. sqr-neg 11.7%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in x around inf 54.8%

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

      1. distribute-lft-out 54.8%

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

      2. fma-def 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in t around 0 32.4%

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

      1. associate-*l/ 39.1%

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

   9. Applied egg-rr 39.1%

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

   if 8.49999999999999996e-64 < t

   1. Initial program 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-commutative 44.1%

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

      3. fma-neg 44.1%

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

      4. sqr-neg 44.1%

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

      5. fma-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in l around 0 94.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-139}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.1e-139) (- t_1) (if (<= t 8.5e-64) (/ (* t (sqrt x)) l) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.1e-139) {
		tmp = -t_1;
	} else if (t <= 8.5e-64) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-2.1d-139)) then
        tmp = -t_1
    else if (t <= 8.5d-64) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.1e-139) {
		tmp = -t_1;
	} else if (t <= 8.5e-64) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -2.1e-139:
		tmp = -t_1
	elif t <= 8.5e-64:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2.1e-139)
		tmp = Float64(-t_1);
	elseif (t <= 8.5e-64)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -2.1e-139)
		tmp = -t_1;
	elseif (t <= 8.5e-64)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.1e-139], (-t$95$1), If[LessEqual[t, 8.5e-64], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.10000000000000008e-139

   1. Initial program 37.4%

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

      1. associate-*l/ 37.4%

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

      2. *-commutative 37.4%

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

      3. fma-neg 37.4%

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

      4. sqr-neg 37.4%

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

      5. fma-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Applied egg-rr 71.5%

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

   6. Taylor expanded in t around -inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. sub-neg 87.0%

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

      3. metadata-eval 87.0%

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

      4. +-commutative 87.0%

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

      5. +-commutative 87.0%

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

   8. Simplified 87.0%

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

   if -2.10000000000000008e-139 < t < 8.49999999999999996e-64

   1. Initial program 11.7%

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

      1. fma-neg 11.8%

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

      2. sqr-neg 11.8%

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

      3. fma-neg 11.7%

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

      4. sqr-neg 11.7%

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

      5. sqr-neg 11.7%

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

      6. sqr-neg 11.7%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in x around inf 54.8%

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

      1. distribute-lft-out 54.8%

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

      2. fma-def 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in t around 0 32.4%

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

      1. associate-*l/ 39.1%

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

   9. Applied egg-rr 39.1%

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

   if 8.49999999999999996e-64 < t

   1. Initial program 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-commutative 44.1%

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

      3. fma-neg 44.1%

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

      4. sqr-neg 44.1%

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

      5. fma-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in l around 0 94.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-139}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 77.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-63}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -9.8e-139)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 4.3e-63)
     (* t (/ (sqrt x) l))
     (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -9.8e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 4.3e-63) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9.8d-139)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 4.3d-63) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -9.8e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 4.3e-63) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -9.8e-139:
		tmp = -1.0 + (1.0 / x)
	elif t <= 4.3e-63:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -9.8e-139)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 4.3e-63)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -9.8e-139)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 4.3e-63)
		tmp = t * (sqrt(x) / l);
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -9.8e-139], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e-63], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.80000000000000063e-139

   1. Initial program 37.4%

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

      1. associate-*l/ 37.4%

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

      2. *-commutative 37.4%

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

      3. fma-neg 37.4%

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

      4. sqr-neg 37.4%

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

      5. fma-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Applied egg-rr 71.5%

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

   6. Taylor expanded in l around 0 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 86.2%

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

   9. Simplified 86.2%

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

   if -9.80000000000000063e-139 < t < 4.2999999999999999e-63

   1. Initial program 11.7%

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

      1. associate-*l/ 11.7%

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

      2. *-commutative 11.7%

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

      3. fma-neg 11.8%

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

      4. sqr-neg 11.8%

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

      5. fma-neg 11.7%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in l around inf 8.3%

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

      1. unpow2 8.3%

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

      2. *-commutative 8.3%

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

      3. associate--l+ 23.1%

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

      4. sub-neg 23.1%

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

      5. metadata-eval 23.1%

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

      6. +-commutative 23.1%

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

      7. sub-neg 23.1%

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

      8. metadata-eval 23.1%

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

      9. +-commutative 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in x around inf 45.0%

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

      1. unpow2 45.0%

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

   9. Simplified 45.0%

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

   10. Taylor expanded in l around 0 39.0%

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

       1. associate-*l/ 39.1%

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

       2. *-lft-identity 39.1%

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

   12. Simplified 39.1%

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

   if 4.2999999999999999e-63 < t

   1. Initial program 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-commutative 44.1%

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

      3. fma-neg 44.1%

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

      4. sqr-neg 44.1%

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

      5. fma-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in l around 0 94.7%

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

   7. Taylor expanded in x around inf 93.1%

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

      1. associate--l+ 93.1%

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

      2. associate-*r/ 93.1%

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

      3. metadata-eval 93.1%

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

      4. unpow2 93.1%

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

   9. Simplified 93.1%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-63}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5: 78.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-63}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.45e-139)
   (+ -1.0 (/ 1.0 x))
   (if (<= t 1.22e-63)
     (/ (* t (sqrt x)) l)
     (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.45e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.22e-63) {
		tmp = (t * sqrt(x)) / l;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.45d-139)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else if (t <= 1.22d-63) then
        tmp = (t * sqrt(x)) / l
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.45e-139) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 1.22e-63) {
		tmp = (t * Math.sqrt(x)) / l;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.45e-139:
		tmp = -1.0 + (1.0 / x)
	elif t <= 1.22e-63:
		tmp = (t * math.sqrt(x)) / l
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.45e-139)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 1.22e-63)
		tmp = Float64(Float64(t * sqrt(x)) / l);
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.45e-139)
		tmp = -1.0 + (1.0 / x);
	elseif (t <= 1.22e-63)
		tmp = (t * sqrt(x)) / l;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.45e-139], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.22e-63], N[(N[(t * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.4499999999999999e-139

   1. Initial program 37.4%

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

      1. associate-*l/ 37.4%

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

      2. *-commutative 37.4%

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

      3. fma-neg 37.4%

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

      4. sqr-neg 37.4%

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

      5. fma-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Applied egg-rr 71.5%

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

   6. Taylor expanded in l around 0 1.6%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 86.2%

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

   9. Simplified 86.2%

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

   if -1.4499999999999999e-139 < t < 1.2199999999999999e-63

   1. Initial program 11.7%

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

      1. fma-neg 11.8%

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

      2. sqr-neg 11.8%

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

      3. fma-neg 11.7%

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

      4. sqr-neg 11.7%

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

      5. sqr-neg 11.7%

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

      6. sqr-neg 11.7%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in x around inf 54.8%

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

      1. distribute-lft-out 54.8%

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

      2. fma-def 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in t around 0 32.4%

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

      1. associate-*l/ 39.1%

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

   9. Applied egg-rr 39.1%

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

   if 1.2199999999999999e-63 < t

   1. Initial program 44.2%

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

      1. associate-*l/ 44.1%

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

      2. *-commutative 44.1%

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

      3. fma-neg 44.1%

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

      4. sqr-neg 44.1%

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

      5. fma-neg 44.1%

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

   3. Simplified 44.1%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in l around 0 94.7%

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

   7. Taylor expanded in x around inf 93.1%

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

      1. associate--l+ 93.1%

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

      2. associate-*r/ 93.1%

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

      3. metadata-eval 93.1%

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

      4. unpow2 93.1%

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

   9. Simplified 93.1%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-139}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-63}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6: 75.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 30.8%

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

      1. associate-*l/ 30.8%

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

      2. *-commutative 30.8%

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

      3. fma-neg 30.8%

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

      4. sqr-neg 30.8%

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

      5. fma-neg 30.8%

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

   3. Simplified 30.8%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in l around 0 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 75.1%

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

   9. Simplified 75.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 35.1%

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

      1. associate-*l/ 35.1%

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

      2. *-commutative 35.1%

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

      3. fma-neg 35.1%

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

      4. sqr-neg 35.1%

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

      5. fma-neg 35.1%

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

   3. Simplified 35.1%

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

   5. Applied egg-rr 68.7%

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

   6. Taylor expanded in l around 0 76.5%

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

   7. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. associate-*r/ 75.4%

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

      3. metadata-eval 75.4%

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

      4. unpow2 75.4%

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

   9. Simplified 75.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7: 75.4% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 30.8%

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

      1. associate-*l/ 30.8%

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

      2. *-commutative 30.8%

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

      3. fma-neg 30.8%

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

      4. sqr-neg 30.8%

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

      5. fma-neg 30.8%

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

   3. Simplified 30.8%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in l around 0 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 75.1%

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

   9. Simplified 75.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 35.1%

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

      1. associate-*l/ 35.1%

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

   3. Simplified 35.1%

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

   4. Taylor expanded in x around inf 73.2%

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

      1. *-commutative 73.2%

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

   6. Simplified 73.2%

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

      1. sqrt-unprod 74.3%

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

      2. metadata-eval 74.3%

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

      3. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 75.8% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ -1.0 (/ 1.0 x)) (- 1.0 (/ 1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 - (1.0d0 / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 30.8%

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

      1. associate-*l/ 30.8%

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

      2. *-commutative 30.8%

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

      3. fma-neg 30.8%

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

      4. sqr-neg 30.8%

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

      5. fma-neg 30.8%

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

   3. Simplified 30.8%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in l around 0 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 75.1%

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

   9. Simplified 75.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 35.1%

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

      1. associate-*l/ 35.1%

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

      2. *-commutative 35.1%

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

      3. fma-neg 35.1%

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

      4. sqr-neg 35.1%

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

      5. fma-neg 35.1%

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

   3. Simplified 35.1%

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

   5. Applied egg-rr 68.7%

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

   6. Taylor expanded in l around 0 76.5%

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

   7. Taylor expanded in x around inf 75.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 9: 75.1% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 30.8%

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

      1. associate-*l/ 30.8%

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

      2. *-commutative 30.8%

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

      3. fma-neg 30.8%

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

      4. sqr-neg 30.8%

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

      5. fma-neg 30.8%

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

   3. Simplified 30.8%

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

   5. Applied egg-rr 62.0%

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

   6. Taylor expanded in l around 0 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

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

      2. rem-square-sqrt 74.1%

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

   9. Simplified 74.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 35.1%

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

      1. associate-*l/ 35.1%

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

   3. Simplified 35.1%

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

   4. Taylor expanded in x around inf 73.2%

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

      1. *-commutative 73.2%

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

   6. Simplified 73.2%

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

      1. sqrt-unprod 74.3%

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

      2. metadata-eval 74.3%

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

      3. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 38.6% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 33.0%

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

   1. associate-*l/ 33.0%

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

   2. *-commutative 33.0%

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

   3. fma-neg 33.0%

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

   4. sqr-neg 33.0%

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

   5. fma-neg 33.0%

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

3. Simplified 33.0%

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

5. Applied egg-rr 65.5%

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

6. Taylor expanded in l around 0 40.3%

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

7. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 0.0%

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

   2. rem-square-sqrt 36.8%

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

9. Simplified 36.8%

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

10. Final simplification 36.8%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023285 
(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)))))