Toniolo and Linder, Equation (7)

Percentage Accurate: 32.6% → 84.8%

Time: 13.4s

Alternatives: 9

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: 32.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 1.28 \cdot 10^{+41}:\\ \;\;\;\;\frac{t\_3}{\sqrt{2 \cdot \left(t\_m \cdot t\_m\right) + \frac{\frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) + \mathsf{fma}\left(2, t\_2, \frac{t\_2}{x}\right)}{x} - t\_2 \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 1.95e-159)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 1.28e+41)
        (/
         t_3
         (sqrt
          (+
           (* 2.0 (* t_m t_m))
           (/
            (-
             (/
              (+
               (fma 2.0 (/ (* t_m t_m) x) (/ (* l l) x))
               (fma 2.0 t_2 (/ t_2 x)))
              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 <= 1.95e-159) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 1.28e+41) {
		tmp = t_3 / sqrt(((2.0 * (t_m * t_m)) + ((((fma(2.0, ((t_m * t_m) / x), ((l * l) / x)) + fma(2.0, t_2, (t_2 / x))) / 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 <= 1.95e-159)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 1.28e+41)
		tmp = Float64(t_3 / sqrt(Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(Float64(Float64(Float64(fma(2.0, Float64(Float64(t_m * t_m) / x), Float64(Float64(l * l) / x)) + fma(2.0, t_2, Float64(t_2 / x))) / 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, 1.95e-159], 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, 1.28e+41], N[(t$95$3 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(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[(2.0 * t$95$2 + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(t$95$2 * -2.0), $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 < 1.94999999999999988e-159

   1. Initial program 32.0%

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

   3. Taylor expanded in 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. Simplified 20.4%

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

   if 1.94999999999999988e-159 < t < 1.27999999999999992e41

   1. Initial program 55.5%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 78.0%

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

   if 1.27999999999999992e41 < t

   1. Initial program 27.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 91.2

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

   5. Simplified 91.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.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. lower-+.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. lower-+.f64 91.3

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

   8. Simplified 91.3%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{+41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t\right) + \frac{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \mathsf{fma}\left(2, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{x} - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\mathsf{fma}\left(2, t\_m \cdot t\_m, \frac{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{t\_2}{x} - t\_2 \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 1.95e-159)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 1.25e-80)
        (/
         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 <= 1.95e-159) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 1.25e-80) {
		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 <= 1.95e-159)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 1.25e-80)
		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, 1.95e-159], 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, 1.25e-80], 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 < 1.94999999999999988e-159

   1. Initial program 32.0%

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

   3. Taylor expanded in 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. Simplified 20.4%

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

   if 1.94999999999999988e-159 < t < 1.25e-80

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. 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. Simplified 79.5%

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

   if 1.25e-80 < t

   1. Initial program 36.2%

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

   3. 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.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. lower-+.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. lower-+.f64 88.5

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

   8. Simplified 88.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot -2\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t_3 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(0.5, \frac{2 \cdot t\_2}{t\_3 \cdot x}, t\_3\right)}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{t\_2}{x} + \mathsf{fma}\left(2, t\_m \cdot t\_m + \frac{t\_m \cdot t\_m}{x}, \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 1.95e-159)
      (/ t_3 (fma 0.5 (/ (* 2.0 t_2) (* t_3 x)) t_3))
      (if (<= t_m 1.25e-80)
        (/
         t_3
         (sqrt
          (+
           (/ t_2 x)
           (fma 2.0 (+ (* t_m t_m) (/ (* t_m t_m) x)) (/ (* 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 = fma(2.0, (t_m * t_m), (l * l));
	double t_3 = t_m * sqrt(2.0);
	double tmp;
	if (t_m <= 1.95e-159) {
		tmp = t_3 / fma(0.5, ((2.0 * t_2) / (t_3 * x)), t_3);
	} else if (t_m <= 1.25e-80) {
		tmp = t_3 / sqrt(((t_2 / x) + fma(2.0, ((t_m * t_m) + ((t_m * t_m) / x)), ((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 = 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 <= 1.95e-159)
		tmp = Float64(t_3 / fma(0.5, Float64(Float64(2.0 * t_2) / Float64(t_3 * x)), t_3));
	elseif (t_m <= 1.25e-80)
		tmp = Float64(t_3 / sqrt(Float64(Float64(t_2 / x) + fma(2.0, Float64(Float64(t_m * t_m) + Float64(Float64(t_m * t_m) / x)), Float64(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[(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, 1.95e-159], 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, 1.25e-80], N[(t$95$3 / N[Sqrt[N[(N[(t$95$2 / x), $MachinePrecision] + N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.94999999999999988e-159

   1. Initial program 32.0%

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

   3. Taylor expanded in 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. Simplified 20.4%

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

   if 1.94999999999999988e-159 < t < 1.25e-80

   1. Initial program 40.8%

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

   3. Taylor expanded in x around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 N/A

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

   5. Simplified 78.1%

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

   if 1.25e-80 < t

   1. Initial program 36.2%

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

   3. 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 88.4

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

   5. Simplified 88.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.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. lower-+.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. lower-+.f64 88.5

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

   8. Simplified 88.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(2, t \cdot t + \frac{t \cdot t}{x}, \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)}{t\_2 \cdot x}, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

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 4.5e-59)
      (/ t_2 (fma 0.5 (/ (* 2.0 (fma 2.0 (* t_m t_m) (* l l))) (* t_2 x)) t_2))
      (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 <= 4.5e-59) {
		tmp = t_2 / fma(0.5, ((2.0 * fma(2.0, (t_m * t_m), (l * l))) / (t_2 * x)), t_2);
	} 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 <= 4.5e-59)
		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));
	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, 4.5e-59], 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], 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 < 4.50000000000000012e-59

   1. Initial program 34.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. 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. Simplified 27.4%

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

   if 4.50000000000000012e-59 < t

   1. Initial program 33.5%

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

   3. Taylor expanded in 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.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. lower-+.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. lower-+.f64 88.7

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

   8. Simplified 88.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\left(t \cdot \sqrt{2}\right) \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \end{array} \]

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 (<= l 8.5e+175) (sqrt (/ (+ x -1.0) (+ x 1.0))) (* t_m (/ (sqrt x) l)))))

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 (l <= 8.5e+175) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (sqrt(x) / l);
	}
	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 (l <= 8.5d+175) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = t_m * (sqrt(x) / l)
    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 (l <= 8.5e+175) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_m * (Math.sqrt(x) / l);
	}
	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 l <= 8.5e+175:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_m * (math.sqrt(x) / l)
	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 (l <= 8.5e+175)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l));
	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 (l <= 8.5e+175)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_m * (sqrt(x) / l);
	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[l, 8.5e+175], 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), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.50000000000000034e175

   1. Initial program 38.0%

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

   3. Taylor expanded in 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 45.2

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

   5. Simplified 45.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.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. lower-+.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. lower-+.f64 45.2

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

   8. Simplified 45.2%

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

   if 8.50000000000000034e175 < 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. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 1.9

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

   5. Simplified 1.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x}} \]
   7. 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. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 54.5

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

   8. Simplified 54.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.7

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-unprod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity 54.8

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

   10. Applied egg-rr 54.8%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 2.6× speedup

Math

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

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 (<= l 4.5e+175) (+ 1.0 (/ -1.0 x)) (* t_m (/ (sqrt x) l)))))

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 (l <= 4.5e+175) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (sqrt(x) / l);
	}
	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 (l <= 4.5d+175) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m * (sqrt(x) / l)
    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 (l <= 4.5e+175) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m * (Math.sqrt(x) / l);
	}
	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 l <= 4.5e+175:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m * (math.sqrt(x) / l)
	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 (l <= 4.5e+175)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m * Float64(sqrt(x) / l));
	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 (l <= 4.5e+175)
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m * (sqrt(x) / l);
	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[l, 4.5e+175], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.49999999999999989e175

   1. Initial program 38.0%

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

   3. Taylor expanded in 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 45.2

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

   5. Simplified 45.2%

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

   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. distribute-neg-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 44.1

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

   8. Simplified 44.1%

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

   if 4.49999999999999989e175 < 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. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 1.9

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

   5. Simplified 1.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x}} \]
   7. 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. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 54.5

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

   8. Simplified 54.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.7

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. sqrt-unprod N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-lft-identity 54.8

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

   10. Applied egg-rr 54.8%

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

4. Final simplification 45.2%

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

Alternative 7: 76.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

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 5.4e-243) (* (sqrt x) (/ t_m l)) (+ 1.0 (/ -1.0 x)))))

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 <= 5.4e-243) {
		tmp = sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= 5.4d-243) then
        tmp = sqrt(x) * (t_m / l)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    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 <= 5.4e-243) {
		tmp = Math.sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	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 <= 5.4e-243:
		tmp = math.sqrt(x) * (t_m / l)
	else:
		tmp = 1.0 + (-1.0 / x)
	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 <= 5.4e-243)
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	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 <= 5.4e-243)
		tmp = sqrt(x) * (t_m / l);
	else
		tmp = 1.0 + (-1.0 / x);
	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, 5.4e-243], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.40000000000000021e-243

   1. Initial program 35.8%

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

   3. Taylor expanded in l around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sub-neg N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 1.6

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

   5. Simplified 1.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{t \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\ell} \cdot \sqrt{x}} \]
   7. 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. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 16.1

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

   8. Simplified 16.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity N/A

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

      17. lower-/.f64 16.2

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

   10. Applied egg-rr 16.2%

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

   if 5.40000000000000021e-243 < t

   1. Initial program 32.5%

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

   3. Taylor expanded in 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 82.0

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

   5. Simplified 82.0%

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

   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. distribute-neg-frac N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 80.1

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

   8. Simplified 80.1%

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

Alternative 8: 75.8% accurate, 5.7× speedup

Math

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

FPCore

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 (/ -1.0 x))))

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 + (-1.0 / x));
}

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 + ((-1.0d0) / x))
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 + (-1.0 / x));
}

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 + (-1.0 / x))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
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 + (-1.0 / x));
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[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. 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. lower-*.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. lower-*.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. lower-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. lower-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. lower-/.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. lower-+.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. lower-+.f64 43.5

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

5. Simplified 43.5%

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

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. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 42.6

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

8. Simplified 42.6%

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

Alternative 9: 75.2% 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 34.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. 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 41.4

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

5. Simplified 41.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
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 42.0

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

7. Applied egg-rr 42.0%

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

Reproduce

herbie shell --seed 2024208 
(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)))))