Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.9%

Time: 25.4s

Alternatives: 9

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: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{t\_3 + t\_3}{{x}^{2}} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))) (t_3 (+ t_2 (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 3.1e-149)
      (* t_m (/ (sqrt (* 2.0 (fma x 0.5 -0.5))) l_m))
      (if (<= t_m 2.1e+40)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+
             (/ (+ t_3 t_3) (pow x 2.0))
             (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x))))
            (/ t_3 x)))))
        (/ 1.0 (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 t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 3.1e-149) {
		tmp = t_m * (sqrt((2.0 * fma(x, 0.5, -0.5))) / l_m);
	} else if (t_m <= 2.1e+40) {
		tmp = t_m * (sqrt(2.0) / sqrt(((((t_3 + t_3) / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = 1.0 / 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)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.1e-149)
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) / l_m));
	elseif (t_m <= 2.1e+40)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(Float64(t_3 + t_3) / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))) + Float64(t_3 / x)))));
	else
		tmp = Float64(1.0 / 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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-149], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+40], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.09999999999999987e-149

   1. Initial program 29.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. associate--l+ 10.5%

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

      3. sub-neg 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. sub-neg 10.5%

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

      7. metadata-eval 10.5%

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

      8. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around 0 19.2%

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

      1. expm1-log1p-u 18.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

      2. expm1-udef 7.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

      3. associate-*r* 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot x - 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

      4. sqrt-unprod 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(0.5 \cdot x - 0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      5. *-commutative 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\color{blue}{x \cdot 0.5} - 0.5\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

      6. fma-neg 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      7. metadata-eval 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

   10. Applied egg-rr 7.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 18.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 19.2%

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

       3. associate-*r/ 21.1%

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

       4. associate-*l/ 21.0%

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

       5. *-commutative 21.0%

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

   12. Simplified 21.0%

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

   if 3.09999999999999987e-149 < t < 2.1000000000000001e40

   1. Initial program 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around -inf 82.8%

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

   if 2.1000000000000001e40 < t

   1. Initial program 35.5%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in t around inf 95.9%

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

   6. Taylor expanded in t around 0 96.0%

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

      1. clear-num 96.1%

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

      2. sqrt-div 96.0%

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

      3. metadata-eval 96.0%

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

      4. +-commutative 96.0%

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

      5. sub-neg 96.0%

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

      6. metadata-eval 96.0%

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

   8. Applied egg-rr 96.0%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 3.1e-149)
      (* t_m (/ (sqrt (* 2.0 (fma x 0.5 -0.5))) l_m))
      (if (<= t_m 1.2e+65)
        (*
         t_m
         (/
          (sqrt 2.0)
          (sqrt
           (+
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
            (/ (+ t_2 (pow l_m 2.0)) x)))))
        (cbrt (pow (/ (+ x -1.0) (+ x 1.0)) 1.5)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3.1e-149) {
		tmp = t_m * (sqrt((2.0 * fma(x, 0.5, -0.5))) / l_m);
	} else if (t_m <= 1.2e+65) {
		tmp = t_m * (sqrt(2.0) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = cbrt(pow(((x + -1.0) / (x + 1.0)), 1.5));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.1e-149)
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) / l_m));
	elseif (t_m <= 1.2e+65)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = cbrt((Float64(Float64(x + -1.0) / Float64(x + 1.0)) ^ 1.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-149], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+65], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.09999999999999987e-149

   1. Initial program 29.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. associate--l+ 10.5%

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

      3. sub-neg 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. sub-neg 10.5%

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

      7. metadata-eval 10.5%

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

      8. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around 0 19.2%

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

      1. expm1-log1p-u 18.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

      2. expm1-udef 7.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

      3. associate-*r* 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot x - 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

      4. sqrt-unprod 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(0.5 \cdot x - 0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      5. *-commutative 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\color{blue}{x \cdot 0.5} - 0.5\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

      6. fma-neg 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      7. metadata-eval 7.9%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

   10. Applied egg-rr 7.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 18.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 19.2%

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

       3. associate-*r/ 21.1%

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

       4. associate-*l/ 21.0%

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

       5. *-commutative 21.0%

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

   12. Simplified 21.0%

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

   if 3.09999999999999987e-149 < t < 1.2000000000000001e65

   1. Initial program 65.0%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x around inf 85.3%

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

   if 1.2000000000000001e65 < t

   1. Initial program 30.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in t around inf 95.6%

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

   6. Taylor expanded in t around 0 95.7%

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

      1. add-cbrt-cube 95.7%

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

      2. add-sqr-sqrt 95.8%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{x - 1}{1 + x}} \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

      3. pow1 95.8%

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

      4. pow1/2 95.8%

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

      5. pow-prod-up 95.8%

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

      6. sub-neg 95.8%

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

      7. metadata-eval 95.8%

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

      8. +-commutative 95.8%

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

      9. metadata-eval 95.8%

         \[\leadsto \sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-116}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.3e-116)
    (* t_m (/ (sqrt (* 2.0 (fma x 0.5 -0.5))) l_m))
    (cbrt (pow (/ (+ x -1.0) (+ x 1.0)) 1.5)))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.3e-116) {
		tmp = t_m * (sqrt((2.0 * fma(x, 0.5, -0.5))) / l_m);
	} else {
		tmp = cbrt(pow(((x + -1.0) / (x + 1.0)), 1.5));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.3e-116)
		tmp = Float64(t_m * Float64(sqrt(Float64(2.0 * fma(x, 0.5, -0.5))) / l_m));
	else
		tmp = cbrt((Float64(Float64(x + -1.0) / Float64(x + 1.0)) ^ 1.5));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.3e-116], N[(t$95$m * N[(N[Sqrt[N[(2.0 * N[(x * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-116

   1. Initial program 30.8%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in l around inf 2.8%

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

      1. *-commutative 2.8%

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

      2. associate--l+ 10.2%

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

      3. sub-neg 10.2%

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

      4. metadata-eval 10.2%

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

      5. +-commutative 10.2%

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

      6. sub-neg 10.2%

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

      7. metadata-eval 10.2%

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

      8. +-commutative 10.2%

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

   7. Simplified 10.2%

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

   8. Taylor expanded in x around 0 18.7%

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

      1. expm1-log1p-u 18.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

      2. expm1-udef 7.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{0.5 \cdot x - 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

      3. associate-*r* 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot x - 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

      4. sqrt-unprod 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(0.5 \cdot x - 0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      5. *-commutative 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\color{blue}{x \cdot 0.5} - 0.5\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

      6. fma-neg 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

      7. metadata-eval 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, \color{blue}{-0.5}\right)} \cdot \frac{t}{\ell}\right)} - 1 \]

   10. Applied egg-rr 7.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 18.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 18.7%

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

       3. associate-*r/ 20.4%

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

       4. associate-*l/ 20.4%

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

       5. *-commutative 20.4%

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

   12. Simplified 20.4%

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

   if 1.3e-116 < t

   1. Initial program 42.1%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in t around inf 87.9%

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

   6. Taylor expanded in t around 0 88.1%

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

      1. add-cbrt-cube 88.1%

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

      2. add-sqr-sqrt 88.1%

         \[\leadsto \sqrt[3]{\color{blue}{\frac{x - 1}{1 + x}} \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

      3. pow1 88.1%

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

      4. pow1/2 88.1%

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

      5. pow-prod-up 88.1%

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

      6. sub-neg 88.1%

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

      7. metadata-eval 88.1%

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

      8. +-commutative 88.1%

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

      9. metadata-eval 88.1%

         \[\leadsto \sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-116}:\\ \;\;\;\;t \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x + -1}{x + 1}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(x \cdot 0.5\right)}}{\frac{l\_m}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.8e-148)
    (/ (sqrt (* 2.0 (* x 0.5))) (/ l_m t_m))
    (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 <= 1.8e-148) {
		tmp = sqrt((2.0 * (x * 0.5))) / (l_m / t_m);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.8d-148) then
        tmp = sqrt((2.0d0 * (x * 0.5d0))) / (l_m / t_m)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.8e-148) {
		tmp = Math.sqrt((2.0 * (x * 0.5))) / (l_m / t_m);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.8e-148:
		tmp = math.sqrt((2.0 * (x * 0.5))) / (l_m / t_m)
	else:
		tmp = math.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 <= 1.8e-148)
		tmp = Float64(sqrt(Float64(2.0 * Float64(x * 0.5))) / Float64(l_m / t_m));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.8e-148)
		tmp = sqrt((2.0 * (x * 0.5))) / (l_m / t_m);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-148], N[(N[Sqrt[N[(2.0 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.7999999999999999e-148

   1. Initial program 29.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. associate--l+ 10.5%

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

      3. sub-neg 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. sub-neg 10.5%

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

      7. metadata-eval 10.5%

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

      8. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around inf 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

       1. associate-*r* 18.8%

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

       2. clear-num 18.8%

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

       3. un-div-inv 18.9%

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

       4. sqrt-unprod 18.9%

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

   12. Applied egg-rr 18.9%

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

   if 1.7999999999999999e-148 < t

   1. Initial program 43.9%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in t around inf 87.6%

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

   6. Taylor expanded in t around 0 87.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(x \cdot 0.5\right)}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot {\left(x + -1\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e-149)
    (* (/ t_m l_m) (pow (+ x -1.0) 0.5))
    (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 <= 3.7e-149) {
		tmp = (t_m / l_m) * pow((x + -1.0), 0.5);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.7d-149) then
        tmp = (t_m / l_m) * ((x + (-1.0d0)) ** 0.5d0)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.7e-149) {
		tmp = (t_m / l_m) * Math.pow((x + -1.0), 0.5);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.7e-149:
		tmp = (t_m / l_m) * math.pow((x + -1.0), 0.5)
	else:
		tmp = math.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 <= 3.7e-149)
		tmp = Float64(Float64(t_m / l_m) * (Float64(x + -1.0) ^ 0.5));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.7e-149)
		tmp = (t_m / l_m) * ((x + -1.0) ^ 0.5);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-149], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Power[N[(x + -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.6999999999999999e-149

   1. Initial program 29.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. associate--l+ 10.5%

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

      3. sub-neg 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. sub-neg 10.5%

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

      7. metadata-eval 10.5%

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

      8. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around 0 19.2%

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

      1. add-sqr-sqrt 9.7%

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

      2. pow2 9.7%

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

      3. associate-*r* 9.7%

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

      4. sqrt-unprod 9.7%

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

      5. *-commutative 9.7%

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

      6. fma-neg 9.7%

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

      7. metadata-eval 9.7%

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

   10. Applied egg-rr 9.7%

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

       1. unpow2 9.7%

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

       2. add-sqr-sqrt 19.2%

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

       3. add-sqr-sqrt 19.2%

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

       4. associate-*l* 19.2%

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

       5. pow1/2 19.2%

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

       6. sqrt-pow1 19.2%

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

       7. metadata-eval 19.2%

          \[\leadsto {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)}} \cdot \frac{t}{\ell}\right) \]

       8. pow1/2 19.2%

          \[\leadsto {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.5}}} \cdot \frac{t}{\ell}\right) \]

       9. sqrt-pow1 19.2%

          \[\leadsto {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot \left(\color{blue}{{\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \frac{t}{\ell}\right) \]

       10. metadata-eval 19.2%

           \[\leadsto {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot \left({\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{\color{blue}{0.25}} \cdot \frac{t}{\ell}\right) \]

   12. Applied egg-rr 19.2%

       \[\leadsto \color{blue}{{\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot \left({\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot \frac{t}{\ell}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 19.2%

          \[\leadsto \color{blue}{\left({\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25}\right) \cdot \frac{t}{\ell}} \]

       2. *-commutative 19.2%

          \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \left({\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25} \cdot {\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{0.25}\right)} \]

       3. pow-sqr 19.2%

          \[\leadsto \frac{t}{\ell} \cdot \color{blue}{{\left(2 \cdot \mathsf{fma}\left(x, 0.5, -0.5\right)\right)}^{\left(2 \cdot 0.25\right)}} \]

       4. fma-udef 19.2%

          \[\leadsto \frac{t}{\ell} \cdot {\left(2 \cdot \color{blue}{\left(x \cdot 0.5 + -0.5\right)}\right)}^{\left(2 \cdot 0.25\right)} \]

       5. distribute-rgt-in 19.2%

          \[\leadsto \frac{t}{\ell} \cdot {\color{blue}{\left(\left(x \cdot 0.5\right) \cdot 2 + -0.5 \cdot 2\right)}}^{\left(2 \cdot 0.25\right)} \]

       6. associate-*l* 19.2%

          \[\leadsto \frac{t}{\ell} \cdot {\left(\color{blue}{x \cdot \left(0.5 \cdot 2\right)} + -0.5 \cdot 2\right)}^{\left(2 \cdot 0.25\right)} \]

       7. metadata-eval 19.2%

          \[\leadsto \frac{t}{\ell} \cdot {\left(x \cdot \color{blue}{1} + -0.5 \cdot 2\right)}^{\left(2 \cdot 0.25\right)} \]

       8. metadata-eval 19.2%

          \[\leadsto \frac{t}{\ell} \cdot {\left(x \cdot 1 + \color{blue}{-1}\right)}^{\left(2 \cdot 0.25\right)} \]

       9. metadata-eval 19.2%

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

   14. Simplified 19.2%

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

   if 3.6999999999999999e-149 < t

   1. Initial program 43.9%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in t around inf 87.6%

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

   6. Taylor expanded in t around 0 87.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{t}{\ell} \cdot {\left(x + -1\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{t\_m}{l\_m} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-149)
    (* (/ t_m l_m) (sqrt 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 <= 3.4e-149) {
		tmp = (t_m / l_m) * sqrt(x);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.4d-149) then
        tmp = (t_m / l_m) * sqrt(x)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.4e-149) {
		tmp = (t_m / l_m) * Math.sqrt(x);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.4e-149:
		tmp = (t_m / l_m) * math.sqrt(x)
	else:
		tmp = math.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 <= 3.4e-149)
		tmp = Float64(Float64(t_m / l_m) * sqrt(x));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.4e-149)
		tmp = (t_m / l_m) * sqrt(x);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-149], N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3999999999999999e-149

   1. Initial program 29.2%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in l around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. associate--l+ 10.5%

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

      3. sub-neg 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. sub-neg 10.5%

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

      7. metadata-eval 10.5%

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

      8. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around inf 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

       1. expm1-log1p-u 18.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

       2. expm1-udef 7.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

       3. associate-*r* 7.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{x \cdot 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

       4. sqrt-unprod 7.9%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(x \cdot 0.5\right)}} \cdot \frac{t}{\ell}\right)} - 1 \]

   12. Applied egg-rr 7.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(x \cdot 0.5\right)} \cdot \frac{t}{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 18.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(x \cdot 0.5\right)} \cdot \frac{t}{\ell}\right)\right)} \]

       2. expm1-log1p 18.9%

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

       3. *-commutative 18.9%

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

       4. *-commutative 18.9%

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

       5. associate-*l* 18.9%

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

       6. metadata-eval 18.9%

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

   14. Simplified 18.9%

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

   if 3.3999999999999999e-149 < t

   1. Initial program 43.9%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in t around inf 87.6%

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

   6. Taylor expanded in t around 0 87.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% 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 1 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 35.1%

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

3. Simplified 35.0%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in t around 0 38.5%

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

7. Final simplification 38.5%

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

Alternative 8: 75.8% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Simplified 35.0%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in x around inf 38.3%

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

7. Final simplification 38.3%

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

Alternative 9: 75.1% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Simplified 35.0%

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

5. Taylor expanded in t around inf 38.4%

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

6. Taylor expanded in x around inf 37.8%

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

7. Final simplification 37.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))))