Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 85.7%

Time: 17.1s

Alternatives: 6

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: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{t\_2}{x} - t\_2 \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma 2.0 (* t_m t_m) (* l l))) (t_3 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-160)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 2.85e+57)
        (/
         t_3
         (sqrt
          (fma
           2.0
           (* t_m t_m)
           (/
            (+
             (fma 2.0 (/ (* t_m t_m) x) (/ (* l l) x))
             (- (/ t_2 x) (* t_2 -2.0)))
            x))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-160) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 2.85e+57) {
		tmp = t_3 / sqrt(fma(2.0, (t_m * t_m), ((fma(2.0, ((t_m * t_m) / x), ((l * l) / x)) + ((t_2 / x) - (t_2 * -2.0))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l * l))
	t_3 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-160)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 2.85e+57)
		tmp = Float64(t_3 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l * l) / x)) + Float64(Float64(t_2 / x) - Float64(t_2 * -2.0))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-160], N[(t$95$3 / N[(0.5 * N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(t$95$3 * x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+57], N[(t$95$3 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / x), $MachinePrecision] - N[(t$95$2 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.5000000000000002e-160

   1. Initial program 23.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. lower-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. Applied rewrites 19.9%

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

   if 9.5000000000000002e-160 < t < 2.8499999999999999e57

   1. Initial program 54.9%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

   5. Applied rewrites 87.6%

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

   if 2.8499999999999999e57 < t

   1. Initial program 20.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right) - -2 \cdot \mathsf{fma}\left(-\ell, \ell + \ell, \ell \cdot \left(\ell + \ell\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0))))
   (*
    t_s
    (if (<= t_m 9.5e-160)
      (/ t_2 (fma 0.5 (/ (* 2.0 (fma 2.0 (* t_m t_m) (* l l))) (* t_2 x)) t_2))
      (if (<= t_m 2.85e+57)
        (/
         t_2
         (sqrt
          (fma
           2.0
           (* t_m t_m)
           (/
            (- (fma l l (* l l)) (* -2.0 (fma (- l) (+ l l) (* l (+ l l)))))
            x))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 9.5e-160) {
		tmp = t_2 / fma(0.5, ((2.0 * fma(2.0, (t_m * t_m), (l * l))) / (t_2 * x)), t_2);
	} else if (t_m <= 2.85e+57) {
		tmp = t_2 / sqrt(fma(2.0, (t_m * t_m), ((fma(l, l, (l * l)) - (-2.0 * fma(-l, (l + l), (l * (l + l))))) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	tmp = 0.0
	if (t_m <= 9.5e-160)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t_m * t_m), Float64(l * l))) / Float64(t_2 * x)), t_2));
	elseif (t_m <= 2.85e+57)
		tmp = Float64(t_2 / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(fma(l, l, Float64(l * l)) - Float64(-2.0 * fma(Float64(-l), Float64(l + l), Float64(l * Float64(l + l))))) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.5e-160], N[(t$95$2 / N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.85e+57], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(N[(l * l + N[(l * l), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[((-l) * N[(l + l), $MachinePrecision] + N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.5000000000000002e-160

   1. Initial program 23.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. lower-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. Applied rewrites 19.9%

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

   if 9.5000000000000002e-160 < t < 2.8499999999999999e57

   1. Initial program 54.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 86.9%

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

   9. Applied rewrites 86.9%

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

   if 2.8499999999999999e57 < t

   1. Initial program 20.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(\ell, \ell, \ell \cdot \ell\right) - -2 \cdot \mathsf{fma}\left(-\ell, \ell + \ell, \ell \cdot \left(\ell + \ell\right)\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\ell \cdot \left(\ell + \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.85e+57)
    (/ (* t_m (sqrt 2.0)) (sqrt (fma 2.0 (* t_m t_m) (/ (* l (+ l l)) x))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.85e+57) {
		tmp = (t_m * sqrt(2.0)) / sqrt(fma(2.0, (t_m * t_m), ((l * (l + l)) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.85e+57)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(fma(2.0, Float64(t_m * t_m), Float64(Float64(l * Float64(l + l)) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.85e+57], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8499999999999999e57

   1. Initial program 31.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 58.4%

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

   9. Applied rewrites 58.7%

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

   if 2.8499999999999999e57 < t

   1. Initial program 20.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 90.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.85 \cdot 10^{+57}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \left(\ell + \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{t\_m \cdot \sqrt{2}}{\sqrt{\ell \cdot \frac{\ell + \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-135)
    (/ (* t_m (sqrt 2.0)) (sqrt (* l (/ (+ l l) x))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.35e-135) {
		tmp = (t_m * sqrt(2.0)) / sqrt((l * ((l + l) / x)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2.35d-135) then
        tmp = (t_m * sqrt(2.0d0)) / sqrt((l * ((l + l) / x)))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 2.35e-135) {
		tmp = (t_m * Math.sqrt(2.0)) / Math.sqrt((l * ((l + l) / x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 2.35e-135:
		tmp = (t_m * math.sqrt(2.0)) / math.sqrt((l * ((l + l) / x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 2.35e-135)
		tmp = Float64(Float64(t_m * sqrt(2.0)) / sqrt(Float64(l * Float64(Float64(l + l) / x))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 2.35e-135)
		tmp = (t_m * sqrt(2.0)) / sqrt((l * ((l + l) / x)));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-135], N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * N[(N[(l + l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.34999999999999988e-135

   1. Initial program 23.0%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 24.3%

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

   10. Applied rewrites 27.5%

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

   if 2.34999999999999988e-135 < t

   1. Initial program 35.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 83.0

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

   5. Applied rewrites 83.0%

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

   7. Applied rewrites 84.3%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \frac{\ell + \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, 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 28.4%

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

3. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 41.2%

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

8. Final simplification 41.2%

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

Alternative 6: 76.1% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 28.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 40.5

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

5. Applied rewrites 40.5%

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

7. Applied rewrites 41.1%

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

Reproduce

herbie shell --seed 2024223 
(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)))))