Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 83.6%

Time: 30.4s

Alternatives: 8

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 0.0017:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot \frac{x + 1}{x + -1}, {l\_m}^{2} \cdot \left(\frac{1}{x + -1} + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-198)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 5.8e-182)
      1.0
      (if (<= t_m 0.0017)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (fma
            2.0
            (* (pow t_m 2.0) (/ (+ x 1.0) (+ x -1.0)))
            (* (pow l_m 2.0) (+ (/ 1.0 (+ x -1.0)) (/ 1.0 x)))))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 4.2e-198) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 5.8e-182) {
		tmp = 1.0;
	} else if (t_m <= 0.0017) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) * ((x + 1.0) / (x + -1.0))), (pow(l_m, 2.0) * ((1.0 / (x + -1.0)) + (1.0 / x))))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 4.2e-198)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 5.8e-182)
		tmp = 1.0;
	elseif (t_m <= 0.0017)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * Float64(Float64(x + 1.0) / Float64(x + -1.0))), Float64((l_m ^ 2.0) * Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(1.0 / x)))))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-198], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 5.8e-182], 1.0, If[LessEqual[t$95$m, 0.0017], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 4.19999999999999986e-198

   1. Initial program 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in t around 0 3.6%

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

      1. associate-/l* 2.9%

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

      2. +-commutative 2.9%

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

      3. sub-neg 2.9%

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

      4. metadata-eval 2.9%

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

      5. fma-neg 2.9%

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

      6. +-commutative 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in x around inf 18.7%

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

      1. cancel-sign-sub-inv 18.7%

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

      2. metadata-eval 18.7%

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

      3. distribute-rgt1-in 18.7%

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

      4. metadata-eval 18.7%

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

   10. Simplified 18.7%

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

   11. Taylor expanded in t around 0 15.9%

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

       1. associate-*l/ 17.8%

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

   13. Simplified 17.8%

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

   if 4.19999999999999986e-198 < t < 5.79999999999999974e-182

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

      1. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 5.79999999999999974e-182 < t < 0.00169999999999999991

   1. Initial program 59.4%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in l around 0 63.8%

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

      1. fma-define 63.8%

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

      2. +-commutative 63.8%

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

      3. associate-*r/ 63.7%

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

      4. sub-neg 63.7%

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

      5. metadata-eval 63.7%

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

      6. +-commutative 63.7%

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

      7. associate--l+ 68.5%

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

      8. sub-neg 68.5%

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

      9. metadata-eval 68.5%

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

      10. +-commutative 68.5%

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

      11. sub-neg 68.5%

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

      12. metadata-eval 68.5%

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

      13. +-commutative 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in x around inf 81.0%

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

   if 0.00169999999999999991 < t

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in l around 0 93.5%

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

      1. associate-*l* 93.5%

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

      2. +-commutative 93.5%

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

      3. sub-neg 93.5%

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

      4. metadata-eval 93.5%

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

      5. +-commutative 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in t around 0 93.7%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0017:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, {\ell}^{2} \cdot \left(\frac{1}{x + -1} + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.45 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 0.00195:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\mathsf{fma}\left(2, {t\_m}^{2} \cdot \frac{x + 1}{x + -1}, \frac{2 \cdot {l\_m}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.5e-198)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 2.45e-181)
      1.0
      (if (<= t_m 0.00195)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (fma
            2.0
            (* (pow t_m 2.0) (/ (+ x 1.0) (+ x -1.0)))
            (/ (* 2.0 (pow l_m 2.0)) x)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.5e-198) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 2.45e-181) {
		tmp = 1.0;
	} else if (t_m <= 0.00195) {
		tmp = sqrt(2.0) * (t_m / sqrt(fma(2.0, (pow(t_m, 2.0) * ((x + 1.0) / (x + -1.0))), ((2.0 * pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.5e-198)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 2.45e-181)
		tmp = 1.0;
	elseif (t_m <= 0.00195)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(fma(2.0, Float64((t_m ^ 2.0) * Float64(Float64(x + 1.0) / Float64(x + -1.0))), Float64(Float64(2.0 * (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.5e-198], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 2.45e-181], 1.0, If[LessEqual[t$95$m, 0.00195], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 2.5e-198

   1. Initial program 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in t around 0 3.6%

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

      1. associate-/l* 2.9%

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

      2. +-commutative 2.9%

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

      3. sub-neg 2.9%

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

      4. metadata-eval 2.9%

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

      5. fma-neg 2.9%

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

      6. +-commutative 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in x around inf 18.7%

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

      1. cancel-sign-sub-inv 18.7%

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

      2. metadata-eval 18.7%

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

      3. distribute-rgt1-in 18.7%

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

      4. metadata-eval 18.7%

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

   10. Simplified 18.7%

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

   11. Taylor expanded in t around 0 15.9%

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

       1. associate-*l/ 17.8%

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

   13. Simplified 17.8%

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

   if 2.5e-198 < t < 2.44999999999999981e-181

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

      1. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.44999999999999981e-181 < t < 0.0019499999999999999

   1. Initial program 59.4%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in l around 0 63.8%

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

      1. fma-define 63.8%

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

      2. +-commutative 63.8%

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

      3. associate-*r/ 63.7%

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

      4. sub-neg 63.7%

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

      5. metadata-eval 63.7%

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

      6. +-commutative 63.7%

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

      7. associate--l+ 68.5%

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

      8. sub-neg 68.5%

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

      9. metadata-eval 68.5%

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

      10. +-commutative 68.5%

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

      11. sub-neg 68.5%

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

      12. metadata-eval 68.5%

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

      13. +-commutative 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in x around inf 80.9%

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

      1. associate-*r/ 80.9%

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

   10. Simplified 80.9%

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

   if 0.0019499999999999999 < t

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in l around 0 93.5%

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

      1. associate-*l* 93.5%

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

      2. +-commutative 93.5%

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

      3. sub-neg 93.5%

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

      4. metadata-eval 93.5%

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

      5. +-commutative 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in t around 0 93.7%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.00195:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(2, {t}^{2} \cdot \frac{x + 1}{x + -1}, \frac{2 \cdot {\ell}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-198}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{x}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.5 \cdot 10^{-175}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-198)
    (/ (* t_m (sqrt x)) l_m)
    (if (<= t_m 1.15e-181)
      1.0
      (if (<= t_m 4.5e-175)
        (* t_m (/ (sqrt x) l_m))
        (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1e-198) {
		tmp = (t_m * sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-181) {
		tmp = 1.0;
	} else if (t_m <= 4.5e-175) {
		tmp = t_m * (sqrt(x) / l_m);
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1d-198) then
        tmp = (t_m * sqrt(x)) / l_m
    else if (t_m <= 1.15d-181) then
        tmp = 1.0d0
    else if (t_m <= 4.5d-175) then
        tmp = t_m * (sqrt(x) / l_m)
    else
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1e-198) {
		tmp = (t_m * Math.sqrt(x)) / l_m;
	} else if (t_m <= 1.15e-181) {
		tmp = 1.0;
	} else if (t_m <= 4.5e-175) {
		tmp = t_m * (Math.sqrt(x) / l_m);
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1e-198:
		tmp = (t_m * math.sqrt(x)) / l_m
	elif t_m <= 1.15e-181:
		tmp = 1.0
	elif t_m <= 4.5e-175:
		tmp = t_m * (math.sqrt(x) / l_m)
	else:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1e-198)
		tmp = Float64(Float64(t_m * sqrt(x)) / l_m);
	elseif (t_m <= 1.15e-181)
		tmp = 1.0;
	elseif (t_m <= 4.5e-175)
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1e-198)
		tmp = (t_m * sqrt(x)) / l_m;
	elseif (t_m <= 1.15e-181)
		tmp = 1.0;
	elseif (t_m <= 4.5e-175)
		tmp = t_m * (sqrt(x) / l_m);
	else
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-198], N[(N[(t$95$m * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.15e-181], 1.0, If[LessEqual[t$95$m, 4.5e-175], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 9.9999999999999991e-199

   1. Initial program 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in t around 0 3.6%

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

      1. associate-/l* 2.9%

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

      2. +-commutative 2.9%

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

      3. sub-neg 2.9%

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

      4. metadata-eval 2.9%

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

      5. fma-neg 2.9%

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

      6. +-commutative 2.9%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in x around inf 18.7%

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

      1. cancel-sign-sub-inv 18.7%

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

      2. metadata-eval 18.7%

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

      3. distribute-rgt1-in 18.7%

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

      4. metadata-eval 18.7%

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

   10. Simplified 18.7%

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

   11. Taylor expanded in t around 0 15.9%

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

       1. associate-*l/ 17.8%

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

   13. Simplified 17.8%

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

   if 9.9999999999999991e-199 < t < 1.14999999999999995e-181

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

      1. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 1.14999999999999995e-181 < t < 4.49999999999999998e-175

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in t around 0 0.0%

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

      1. associate-/l* 0.0%

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

      2. +-commutative 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. fma-neg 0.0%

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

      6. +-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 31.7%

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

      1. cancel-sign-sub-inv 31.7%

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

      2. metadata-eval 31.7%

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

      3. distribute-rgt1-in 31.7%

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

      4. metadata-eval 31.7%

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

   10. Simplified 31.7%

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

   11. Taylor expanded in t around 0 31.7%

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

   12. Taylor expanded in x around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. associate-*l/ 0.0%

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

       3. associate-/l* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 92.6%

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

       7. neg-mul-1 92.6%

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

       8. distribute-lft-neg-in 92.6%

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

       9. remove-double-neg 92.6%

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

   14. Simplified 92.6%

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

   if 4.49999999999999998e-175 < t

   1. Initial program 45.5%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in l around 0 87.3%

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

      1. associate-*l* 87.3%

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

      2. +-commutative 87.3%

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

      3. sub-neg 87.3%

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

      4. metadata-eval 87.3%

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

      5. +-commutative 87.3%

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

   7. Simplified 87.3%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

      3. sqrt-unprod 87.4%

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

      4. +-commutative 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. unpow-1 87.4%

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

       2. associate-/l* 87.4%

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

   11. Simplified 87.4%

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

       1. un-div-inv 87.4%

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

       2. *-commutative 87.4%

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

       3. associate-/r* 87.3%

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

   13. Applied egg-rr 87.3%

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}{t}}}{t}} \]
   14. Step-by-step derivation

       1. associate-/l/ 87.4%

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

       2. *-un-lft-identity 87.4%

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

       3. times-frac 87.2%

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

       4. associate-/r/ 87.3%

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

       5. sqrt-undiv 87.3%

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

   15. Applied egg-rr 87.3%

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

   16. Taylor expanded in x around inf 86.4%

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

       1. associate--l+ 86.4%

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

       2. unpow2 86.4%

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

       3. metadata-eval 86.4%

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

       4. metadata-eval 86.4%

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

       5. metadata-eval 86.4%

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

       6. rem-square-sqrt 0.0%

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

       7. unpow2 0.0%

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

       8. associate-/l/ 0.0%

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

       9. associate-*r/ 0.0%

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

       10. div-sub 0.0%

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

   18. Simplified 86.4%

       \[\leadsto \color{blue}{1 + \frac{\frac{0.5}{x} + -1}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-198}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-198}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.4 \cdot 10^{-175}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (sqrt x) l_m))))
   (*
    t_s
    (if (<= t_m 3.9e-198)
      t_2
      (if (<= t_m 3.3e-182)
        1.0
        (if (<= t_m 4.4e-175) t_2 (+ 1.0 (/ (+ -1.0 (/ 0.5 x)) x))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.9e-198) {
		tmp = t_2;
	} else if (t_m <= 3.3e-182) {
		tmp = 1.0;
	} else if (t_m <= 4.4e-175) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sqrt(x) / l_m)
    if (t_m <= 3.9d-198) then
        tmp = t_2
    else if (t_m <= 3.3d-182) then
        tmp = 1.0d0
    else if (t_m <= 4.4d-175) then
        tmp = t_2
    else
        tmp = 1.0d0 + (((-1.0d0) + (0.5d0 / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * (Math.sqrt(x) / l_m);
	double tmp;
	if (t_m <= 3.9e-198) {
		tmp = t_2;
	} else if (t_m <= 3.3e-182) {
		tmp = 1.0;
	} else if (t_m <= 4.4e-175) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * (math.sqrt(x) / l_m)
	tmp = 0
	if t_m <= 3.9e-198:
		tmp = t_2
	elif t_m <= 3.3e-182:
		tmp = 1.0
	elif t_m <= 4.4e-175:
		tmp = t_2
	else:
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * Float64(sqrt(x) / l_m))
	tmp = 0.0
	if (t_m <= 3.9e-198)
		tmp = t_2;
	elseif (t_m <= 3.3e-182)
		tmp = 1.0;
	elseif (t_m <= 4.4e-175)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * (sqrt(x) / l_m);
	tmp = 0.0;
	if (t_m <= 3.9e-198)
		tmp = t_2;
	elseif (t_m <= 3.3e-182)
		tmp = 1.0;
	elseif (t_m <= 4.4e-175)
		tmp = t_2;
	else
		tmp = 1.0 + ((-1.0 + (0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.9e-198], t$95$2, If[LessEqual[t$95$m, 3.3e-182], 1.0, If[LessEqual[t$95$m, 4.4e-175], t$95$2, N[(1.0 + N[(N[(-1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.8999999999999999e-198 or 3.29999999999999996e-182 < t < 4.4e-175

   1. Initial program 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in t around 0 3.6%

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

      1. associate-/l* 2.8%

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

      2. +-commutative 2.8%

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

      3. sub-neg 2.8%

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

      4. metadata-eval 2.8%

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

      5. fma-neg 2.8%

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

      6. +-commutative 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in x around inf 19.0%

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

      1. cancel-sign-sub-inv 19.0%

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

      2. metadata-eval 19.0%

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

      3. distribute-rgt1-in 19.0%

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

      4. metadata-eval 19.0%

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

   10. Simplified 19.0%

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

   11. Taylor expanded in t around 0 16.2%

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

   12. Taylor expanded in x around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. associate-*l/ 0.0%

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

       3. associate-/l* 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 19.3%

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

       7. neg-mul-1 19.3%

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

       8. distribute-lft-neg-in 19.3%

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

       9. remove-double-neg 19.3%

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

   14. Simplified 19.3%

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

   if 3.8999999999999999e-198 < t < 3.29999999999999996e-182

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in l around 0 100.0%

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

      1. associate-*l* 100.0%

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

      2. +-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 4.4e-175 < t

   1. Initial program 45.5%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in l around 0 87.3%

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

      1. associate-*l* 87.3%

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

      2. +-commutative 87.3%

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

      3. sub-neg 87.3%

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

      4. metadata-eval 87.3%

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

      5. +-commutative 87.3%

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

   7. Simplified 87.3%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

      3. sqrt-unprod 87.4%

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

      4. +-commutative 87.4%

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

   9. Applied egg-rr 87.4%

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

       1. unpow-1 87.4%

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

       2. associate-/l* 87.4%

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

   11. Simplified 87.4%

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

       1. un-div-inv 87.4%

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

       2. *-commutative 87.4%

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

       3. associate-/r* 87.3%

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

   13. Applied egg-rr 87.3%

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}{t}}}{t}} \]
   14. Step-by-step derivation

       1. associate-/l/ 87.4%

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

       2. *-un-lft-identity 87.4%

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

       3. times-frac 87.2%

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

       4. associate-/r/ 87.3%

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

       5. sqrt-undiv 87.3%

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

   15. Applied egg-rr 87.3%

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

   16. Taylor expanded in x around inf 86.4%

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

       1. associate--l+ 86.4%

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

       2. unpow2 86.4%

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

       3. metadata-eval 86.4%

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

       4. metadata-eval 86.4%

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

       5. metadata-eval 86.4%

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

       6. rem-square-sqrt 0.0%

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

       7. unpow2 0.0%

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

       8. associate-/l/ 0.0%

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

       9. associate-*r/ 0.0%

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

       10. div-sub 0.0%

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

   18. Simplified 86.4%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-198}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 + \frac{0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in l around 0 40.6%

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

   1. associate-*l* 40.6%

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

   2. +-commutative 40.6%

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

   3. sub-neg 40.6%

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

   4. metadata-eval 40.6%

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

   5. +-commutative 40.6%

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

7. Simplified 40.6%

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

8. Taylor expanded in t around 0 40.7%

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

9. Final simplification 40.7%

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

Alternative 6: 76.6% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in l around 0 40.6%

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

   1. associate-*l* 40.6%

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

   2. +-commutative 40.6%

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

   3. sub-neg 40.6%

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

   4. metadata-eval 40.6%

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

   5. +-commutative 40.6%

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

7. Simplified 40.6%

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

   1. clear-num 40.7%

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

   2. inv-pow 40.7%

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

   3. sqrt-unprod 40.7%

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

   4. +-commutative 40.7%

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

9. Applied egg-rr 40.7%

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

    1. unpow-1 40.7%

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

    2. associate-/l* 40.6%

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

11. Simplified 40.6%

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

    1. un-div-inv 40.7%

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

    2. *-commutative 40.7%

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

    3. associate-/r* 40.6%

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

13. Applied egg-rr 40.6%

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{x + 1}{x + -1}}}{t}}}{t}} \]
14. Step-by-step derivation

    1. associate-/l/ 40.7%

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

    2. *-un-lft-identity 40.7%

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

    3. times-frac 40.5%

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

    4. associate-/r/ 40.6%

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

    5. sqrt-undiv 40.6%

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

15. Applied egg-rr 40.6%

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

16. Taylor expanded in x around inf 40.2%

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

    1. associate--l+ 40.2%

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

    2. unpow2 40.2%

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

    3. metadata-eval 40.2%

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

    4. metadata-eval 40.2%

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

    5. metadata-eval 40.2%

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

    6. rem-square-sqrt 0.0%

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

    7. unpow2 0.0%

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

    8. associate-/l/ 0.0%

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

    9. associate-*r/ 0.0%

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

    10. div-sub 0.0%

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

18. Simplified 40.2%

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

19. Final simplification 40.2%

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

Alternative 7: 76.5% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in l around 0 40.6%

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

   1. associate-*l* 40.6%

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

   2. +-commutative 40.6%

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

   3. sub-neg 40.6%

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

   4. metadata-eval 40.6%

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

   5. +-commutative 40.6%

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

7. Simplified 40.6%

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

8. Taylor expanded in x around inf 39.9%

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

9. Final simplification 39.9%

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

Alternative 8: 75.9% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 35.2%

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

5. Taylor expanded in l around 0 40.6%

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

   1. associate-*l* 40.6%

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

   2. +-commutative 40.6%

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

   3. sub-neg 40.6%

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

   4. metadata-eval 40.6%

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

   5. +-commutative 40.6%

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

7. Simplified 40.6%

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

8. Taylor expanded in x around inf 39.2%

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

Reproduce

herbie shell --seed 2024107 
(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)))))