Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 80.9%

Time: 13.5s

Alternatives: 5

Speedup: 85.0×

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: 33.9% 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: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\mathsf{fma}\left(l\_m, \frac{l\_m}{x \cdot t\_2}, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \end{array} \]

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 2.0))))
   (*
    t_s
    (if (<= l_m 1.05e+194)
      (* (sqrt 2.0) (/ t_m (fma l_m (/ l_m (* x t_2)) t_2)))
      (* t_m (/ (sqrt x) l_m))))))

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(2.0);
	double tmp;
	if (l_m <= 1.05e+194) {
		tmp = sqrt(2.0) * (t_m / fma(l_m, (l_m / (x * t_2)), t_2));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	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 * sqrt(2.0))
	tmp = 0.0
	if (l_m <= 1.05e+194)
		tmp = Float64(sqrt(2.0) * Float64(t_m / fma(l_m, Float64(l_m / Float64(x * t_2)), t_2)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	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[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 1.05e+194], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l$95$m * N[(l$95$m / N[(x * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.05000000000000008e194

   1. Initial program 36.4%

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

   3. Taylor expanded in x around inf

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 29.2%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.5

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

   8. Simplified 43.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 46.2%

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

   if 1.05000000000000008e194 < l

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-lowering-+.f64 48.3

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

   7. Simplified 48.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 89.4

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

   10. Simplified 89.4%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. sqrt-unprod N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. *-lowering-*.f64 90.0

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

   12. Applied egg-rr 90.0%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(\ell, \frac{\ell}{x \cdot \left(t \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 6.4 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{l\_m}\\ \end{array} \end{array} \]

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 (<= l_m 6.4e+197)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* t_m (/ (sqrt x) l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 6.4e+197) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 6.4d+197) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(x) / l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 6.4e+197) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 6.4e+197:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 6.4e+197)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 6.4e+197)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(x) / l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 6.4e+197], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.3999999999999997e197

   1. Initial program 36.4%

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

   3. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 44.4

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

   5. Simplified 44.4%

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

   6. Taylor expanded in t around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 44.4

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

   8. Simplified 44.4%

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

   if 6.3999999999999997e197 < l

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. div-inv N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in l around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-lowering-+.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-lowering-+.f64 48.3

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

   7. Simplified 48.3%

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

   8. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 89.4

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

   10. Simplified 89.4%

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. sqrt-lowering-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. sqrt-unprod N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. *-lowering-*.f64 90.0

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

   12. Applied egg-rr 90.0%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.4 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 3.0× speedup

Math

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

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 33.7%

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

3. Taylor expanded in l around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-lowering-+.f64 42.1

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

5. Simplified 42.1%

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

6. Taylor expanded in t around 0

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

   1. sqrt-lowering-sqrt.f64 N/A

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

   2. /-lowering-/.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-lowering-+.f64 N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 42.1

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

8. Simplified 42.1%

   \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x + 1}}} \]
9. Add Preprocessing

Alternative 4: 76.7% accurate, 5.7× speedup

Math

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

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 33.7%

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

3. Taylor expanded in l around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-lowering-+.f64 42.1

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

5. Simplified 42.1%

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

6. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 42.0

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

8. Simplified 42.0%

   \[\leadsto \color{blue}{1 + \frac{-1}{x}} \]
9. Add Preprocessing

Alternative 5: 76.0% accurate, 85.0× 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 33.7%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{1}{2}}} \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{1}{2}} \]

   4. sqrt-lowering-sqrt.f64 41.2

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

5. Simplified 41.2%

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

   1. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{1}{2}}} \]

   2. metadata-eval N/A

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

   3. metadata-eval 41.8

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

7. Applied egg-rr 41.8%

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

Reproduce

herbie shell --seed 2024198 
(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)))))