Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 84.8%

Time: 25.5s

Alternatives: 8

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := t\_3 - -1 \cdot t\_3\\ t_5 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_5}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_5}{\sqrt{-1 \cdot \frac{\left(-1 \cdot t\_4 + -1 \cdot \frac{t\_3}{x}\right) - \left(2 \cdot \frac{{t\_m}^{2}}{x} + \frac{{\ell}^{2}}{x}\right)}{x} + t\_2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (- t_3 (* -1.0 t_3)))
        (t_5 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.7e-158)
      (/ t_5 (+ (* 0.5 (/ t_4 (* t_m (* x (sqrt 2.0))))) (* t_m (sqrt 2.0))))
      (if (<= t_m 5e-9)
        (/
         t_5
         (sqrt
          (+
           (*
            -1.0
            (/
             (-
              (+ (* -1.0 t_4) (* -1.0 (/ t_3 x)))
              (+ (* 2.0 (/ (pow t_m 2.0) x)) (/ (pow l 2.0) x)))
             x))
           t_2)))
        (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5))))))

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 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = t_3 - (-1.0 * t_3);
	double t_5 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.7e-158) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0)));
	} else if (t_m <= 5e-9) {
		tmp = t_5 / sqrt(((-1.0 * ((((-1.0 * t_4) + (-1.0 * (t_3 / x))) - ((2.0 * (pow(t_m, 2.0) / x)) + (pow(l, 2.0) / x))) / x)) + t_2));
	} else {
		tmp = pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = t_3 - ((-1.0d0) * t_3)
    t_5 = sqrt(2.0d0) * t_m
    if (t_m <= 2.7d-158) then
        tmp = t_5 / ((0.5d0 * (t_4 / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0)))
    else if (t_m <= 5d-9) then
        tmp = t_5 / sqrt((((-1.0d0) * (((((-1.0d0) * t_4) + ((-1.0d0) * (t_3 / x))) - ((2.0d0 * ((t_m ** 2.0d0) / x)) + ((l ** 2.0d0) / x))) / x)) + t_2))
    else
        tmp = ((1.0d0 + x) / (x + (-1.0d0))) ** (-0.5d0)
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = t_3 - (-1.0 * t_3);
	double t_5 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.7e-158) {
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0)));
	} else if (t_m <= 5e-9) {
		tmp = t_5 / Math.sqrt(((-1.0 * ((((-1.0 * t_4) + (-1.0 * (t_3 / x))) - ((2.0 * (Math.pow(t_m, 2.0) / x)) + (Math.pow(l, 2.0) / x))) / x)) + t_2));
	} else {
		tmp = Math.pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = t_3 - (-1.0 * t_3)
	t_5 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 2.7e-158:
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0)))
	elif t_m <= 5e-9:
		tmp = t_5 / math.sqrt(((-1.0 * ((((-1.0 * t_4) + (-1.0 * (t_3 / x))) - ((2.0 * (math.pow(t_m, 2.0) / x)) + (math.pow(l, 2.0) / x))) / x)) + t_2))
	else:
		tmp = math.pow(((1.0 + x) / (x + -1.0)), -0.5)
	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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(t_3 - Float64(-1.0 * t_3))
	t_5 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.7e-158)
		tmp = Float64(t_5 / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 5e-9)
		tmp = Float64(t_5 / sqrt(Float64(Float64(-1.0 * Float64(Float64(Float64(Float64(-1.0 * t_4) + Float64(-1.0 * Float64(t_3 / x))) - Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64((l ^ 2.0) / x))) / x)) + t_2)));
	else
		tmp = Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5;
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = t_3 - (-1.0 * t_3);
	t_5 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 2.7e-158)
		tmp = t_5 / ((0.5 * (t_4 / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0)));
	elseif (t_m <= 5e-9)
		tmp = t_5 / sqrt(((-1.0 * ((((-1.0 * t_4) + (-1.0 * (t_3 / x))) - ((2.0 * ((t_m ^ 2.0) / x)) + ((l ^ 2.0) / x))) / x)) + t_2));
	else
		tmp = ((1.0 + x) / (x + -1.0)) ^ -0.5;
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.7e-158], N[(t$95$5 / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e-9], N[(t$95$5 / N[Sqrt[N[(N[(-1.0 * N[(N[(N[(N[(-1.0 * t$95$4), $MachinePrecision] + N[(-1.0 * N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.6999999999999998e-158

   1. Initial program 31.0%

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

   3. Taylor expanded in x around inf 13.2%

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

   if 2.6999999999999998e-158 < t < 5.0000000000000001e-9

   1. Initial program 44.7%

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

   3. Taylor expanded in x around -inf 73.0%

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

   if 5.0000000000000001e-9 < t

   1. Initial program 41.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around inf 91.4%

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

   6. Taylor expanded in t around 0 91.7%

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

      1. clear-num 91.7%

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

      2. sub-neg 91.7%

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

      3. metadata-eval 91.7%

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

      4. sqrt-div 91.7%

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

      5. metadata-eval 91.7%

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

      6. +-commutative 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. pow1/2 91.7%

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

      2. pow-flip 91.7%

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

      3. +-commutative 91.7%

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

      4. metadata-eval 91.7%

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

   10. Applied egg-rr 91.7%

       \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.02e-160)
      (/
       1.0
       (+
        1.0
        (+ (/ 0.5 (pow x 2.0)) (+ (/ 1.0 x) (* 0.5 (/ 1.0 (pow x 3.0)))))))
      (if (<= t_m 2.2e-10)
        (*
         (sqrt 2.0)
         (/
          t_m
          (sqrt
           (-
            (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
            (* -1.0 (/ (+ t_2 (pow l 2.0)) x))))))
        (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5))))))

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 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.02e-160) {
		tmp = 1.0 / (1.0 + ((0.5 / pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / pow(x, 3.0))))));
	} else if (t_m <= 2.2e-10) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * ((t_2 + pow(l, 2.0)) / x)))));
	} else {
		tmp = pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1.02d-160) then
        tmp = 1.0d0 / (1.0d0 + ((0.5d0 / (x ** 2.0d0)) + ((1.0d0 / x) + (0.5d0 * (1.0d0 / (x ** 3.0d0))))))
    else if (t_m <= 2.2d-10) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * ((t_2 + (l ** 2.0d0)) / x)))))
    else
        tmp = ((1.0d0 + x) / (x + (-1.0d0))) ** (-0.5d0)
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.02e-160) {
		tmp = 1.0 / (1.0 + ((0.5 / Math.pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / Math.pow(x, 3.0))))));
	} else if (t_m <= 2.2e-10) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + Math.pow(l, 2.0)) / x)))));
	} else {
		tmp = Math.pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 1.02e-160:
		tmp = 1.0 / (1.0 + ((0.5 / math.pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / math.pow(x, 3.0))))))
	elif t_m <= 2.2e-10:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + math.pow(l, 2.0)) / x)))))
	else:
		tmp = math.pow(((1.0 + x) / (x + -1.0)), -0.5)
	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(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.02e-160)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(1.0 / x) + Float64(0.5 * Float64(1.0 / (x ^ 3.0)))))));
	elseif (t_m <= 2.2e-10)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(Float64(t_2 + (l ^ 2.0)) / x))))));
	else
		tmp = Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5;
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.02e-160)
		tmp = 1.0 / (1.0 + ((0.5 / (x ^ 2.0)) + ((1.0 / x) + (0.5 * (1.0 / (x ^ 3.0))))));
	elseif (t_m <= 2.2e-10)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * ((t_2 + (l ^ 2.0)) / x)))));
	else
		tmp = ((1.0 + x) / (x + -1.0)) ^ -0.5;
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.02e-160], N[(1.0 / N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-10], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.0200000000000001e-160

   1. Initial program 31.2%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in t around inf 8.1%

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

   6. Taylor expanded in t around 0 8.1%

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

      1. clear-num 8.1%

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

      2. sub-neg 8.1%

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

      3. metadata-eval 8.1%

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

      4. sqrt-div 8.1%

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

      5. metadata-eval 8.1%

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

      6. +-commutative 8.1%

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

   8. Applied egg-rr 8.1%

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

   9. Taylor expanded in x around inf 8.1%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)}} \]

   if 1.0200000000000001e-160 < t < 2.1999999999999999e-10

   1. Initial program 43.3%

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

   3. Simplified 12.6%

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

   5. Taylor expanded in x around inf 73.7%

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

   if 2.1999999999999999e-10 < t

   1. Initial program 41.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around inf 91.4%

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

   6. Taylor expanded in t around 0 91.7%

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

      1. clear-num 91.7%

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

      2. sub-neg 91.7%

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

      3. metadata-eval 91.7%

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

      4. sqrt-div 91.7%

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

      5. metadata-eval 91.7%

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

      6. +-commutative 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. pow1/2 91.7%

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

      2. pow-flip 91.7%

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

      3. +-commutative 91.7%

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

      4. metadata-eval 91.7%

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

   10. Applied egg-rr 91.7%

       \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_2 + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.55e-159)
      (/
       1.0
       (+
        1.0
        (+ (/ 0.5 (pow x 2.0)) (+ (/ 1.0 x) (* 0.5 (/ 1.0 (pow x 3.0)))))))
      (if (<= t_m 6e-9)
        (/
         (* (sqrt 2.0) t_m)
         (sqrt
          (-
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
           (* -1.0 (/ (+ t_2 (pow l 2.0)) x)))))
        (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5))))))

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 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.55e-159) {
		tmp = 1.0 / (1.0 + ((0.5 / pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / pow(x, 3.0))))));
	} else if (t_m <= 6e-9) {
		tmp = (sqrt(2.0) * t_m) / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * ((t_2 + pow(l, 2.0)) / x))));
	} else {
		tmp = pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 1.55d-159) then
        tmp = 1.0d0 / (1.0d0 + ((0.5d0 / (x ** 2.0d0)) + ((1.0d0 / x) + (0.5d0 * (1.0d0 / (x ** 3.0d0))))))
    else if (t_m <= 6d-9) then
        tmp = (sqrt(2.0d0) * t_m) / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * ((t_2 + (l ** 2.0d0)) / x))))
    else
        tmp = ((1.0d0 + x) / (x + (-1.0d0))) ** (-0.5d0)
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 1.55e-159) {
		tmp = 1.0 / (1.0 + ((0.5 / Math.pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / Math.pow(x, 3.0))))));
	} else if (t_m <= 6e-9) {
		tmp = (Math.sqrt(2.0) * t_m) / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + Math.pow(l, 2.0)) / x))));
	} else {
		tmp = Math.pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 1.55e-159:
		tmp = 1.0 / (1.0 + ((0.5 / math.pow(x, 2.0)) + ((1.0 / x) + (0.5 * (1.0 / math.pow(x, 3.0))))))
	elif t_m <= 6e-9:
		tmp = (math.sqrt(2.0) * t_m) / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * ((t_2 + math.pow(l, 2.0)) / x))))
	else:
		tmp = math.pow(((1.0 + x) / (x + -1.0)), -0.5)
	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(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.55e-159)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(1.0 / x) + Float64(0.5 * Float64(1.0 / (x ^ 3.0)))))));
	elseif (t_m <= 6e-9)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(Float64(t_2 + (l ^ 2.0)) / x)))));
	else
		tmp = Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5;
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 1.55e-159)
		tmp = 1.0 / (1.0 + ((0.5 / (x ^ 2.0)) + ((1.0 / x) + (0.5 * (1.0 / (x ^ 3.0))))));
	elseif (t_m <= 6e-9)
		tmp = (sqrt(2.0) * t_m) / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * ((t_2 + (l ^ 2.0)) / x))));
	else
		tmp = ((1.0 + x) / (x + -1.0)) ^ -0.5;
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-159], N[(1.0 / N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e-9], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.55e-159

   1. Initial program 31.2%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in t around inf 8.1%

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

   6. Taylor expanded in t around 0 8.1%

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

      1. clear-num 8.1%

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

      2. sub-neg 8.1%

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

      3. metadata-eval 8.1%

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

      4. sqrt-div 8.1%

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

      5. metadata-eval 8.1%

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

      6. +-commutative 8.1%

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

   8. Applied egg-rr 8.1%

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

   9. Taylor expanded in x around inf 8.1%

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)}} \]

   if 1.55e-159 < t < 5.99999999999999996e-9

   1. Initial program 43.3%

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

   3. Taylor expanded in x around inf 73.7%

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

   if 5.99999999999999996e-9 < t

   1. Initial program 41.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around inf 91.4%

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

   6. Taylor expanded in t around 0 91.7%

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

      1. clear-num 91.7%

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

      2. sub-neg 91.7%

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

      3. metadata-eval 91.7%

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

      4. sqrt-div 91.7%

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

      5. metadata-eval 91.7%

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

      6. +-commutative 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. pow1/2 91.7%

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

      2. pow-flip 91.7%

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

      3. +-commutative 91.7%

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

      4. metadata-eval 91.7%

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

   10. Applied egg-rr 91.7%

       \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {\ell}^{2}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-158}:\\ \;\;\;\;\frac{t\_4}{0.5 \cdot \frac{t\_3 - -1 \cdot t\_3}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

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 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l 2.0)))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 2.9e-158)
      (/
       t_4
       (+
        (* 0.5 (/ (- t_3 (* -1.0 t_3)) (* t_m (* x (sqrt 2.0)))))
        (* t_m (sqrt 2.0))))
      (if (<= t_m 3.8e-8)
        (/
         t_4
         (sqrt
          (-
           (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l 2.0) x)))
           (* -1.0 (/ t_3 x)))))
        (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5))))))

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 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l, 2.0);
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.9e-158) {
		tmp = t_4 / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0)));
	} else if (t_m <= 3.8e-8) {
		tmp = t_4 / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))));
	} else {
		tmp = pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    t_3 = t_2 + (l ** 2.0d0)
    t_4 = sqrt(2.0d0) * t_m
    if (t_m <= 2.9d-158) then
        tmp = t_4 / ((0.5d0 * ((t_3 - ((-1.0d0) * t_3)) / (t_m * (x * sqrt(2.0d0))))) + (t_m * sqrt(2.0d0)))
    else if (t_m <= 3.8d-8) then
        tmp = t_4 / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l ** 2.0d0) / x))) - ((-1.0d0) * (t_3 / x))))
    else
        tmp = ((1.0d0 + x) / (x + (-1.0d0))) ** (-0.5d0)
    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 t_2 = 2.0 * Math.pow(t_m, 2.0);
	double t_3 = t_2 + Math.pow(l, 2.0);
	double t_4 = Math.sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 2.9e-158) {
		tmp = t_4 / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * Math.sqrt(2.0))))) + (t_m * Math.sqrt(2.0)));
	} else if (t_m <= 3.8e-8) {
		tmp = t_4 / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))));
	} else {
		tmp = Math.pow(((1.0 + x) / (x + -1.0)), -0.5);
	}
	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):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	t_3 = t_2 + math.pow(l, 2.0)
	t_4 = math.sqrt(2.0) * t_m
	tmp = 0
	if t_m <= 2.9e-158:
		tmp = t_4 / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * math.sqrt(2.0))))) + (t_m * math.sqrt(2.0)))
	elif t_m <= 3.8e-8:
		tmp = t_4 / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l, 2.0) / x))) - (-1.0 * (t_3 / x))))
	else:
		tmp = math.pow(((1.0 + x) / (x + -1.0)), -0.5)
	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(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l ^ 2.0))
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 2.9e-158)
		tmp = Float64(t_4 / Float64(Float64(0.5 * Float64(Float64(t_3 - Float64(-1.0 * t_3)) / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0))));
	elseif (t_m <= 3.8e-8)
		tmp = Float64(t_4 / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l ^ 2.0) / x))) - Float64(-1.0 * Float64(t_3 / x)))));
	else
		tmp = Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5;
	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)
	t_2 = 2.0 * (t_m ^ 2.0);
	t_3 = t_2 + (l ^ 2.0);
	t_4 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (t_m <= 2.9e-158)
		tmp = t_4 / ((0.5 * ((t_3 - (-1.0 * t_3)) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0)));
	elseif (t_m <= 3.8e-8)
		tmp = t_4 / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l ^ 2.0) / x))) - (-1.0 * (t_3 / x))));
	else
		tmp = ((1.0 + x) / (x + -1.0)) ^ -0.5;
	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_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-158], N[(t$95$4 / N[(N[(0.5 * N[(N[(t$95$3 - N[(-1.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.8e-8], N[(t$95$4 / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8999999999999998e-158

   1. Initial program 31.0%

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

   3. Taylor expanded in x around inf 13.2%

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

   if 2.8999999999999998e-158 < t < 3.80000000000000028e-8

   1. Initial program 44.7%

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

   3. Taylor expanded in x around inf 72.9%

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

   if 3.80000000000000028e-8 < t

   1. Initial program 41.9%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around inf 91.4%

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

   6. Taylor expanded in t around 0 91.7%

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

      1. clear-num 91.7%

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

      2. sub-neg 91.7%

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

      3. metadata-eval 91.7%

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

      4. sqrt-div 91.7%

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

      5. metadata-eval 91.7%

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

      6. +-commutative 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. pow1/2 91.7%

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

      2. pow-flip 91.7%

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

      3. +-commutative 91.7%

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

      4. metadata-eval 91.7%

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

   10. Applied egg-rr 91.7%

       \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.5% accurate, 2.1× speedup

Math

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

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 (pow (/ (+ 1.0 x) (+ x -1.0)) -0.5)))

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 * pow(((1.0 + x) / (x + -1.0)), -0.5);
}

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 + x) / (x + (-1.0d0))) ** (-0.5d0))
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.pow(((1.0 + x) / (x + -1.0)), -0.5);
}

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.pow(((1.0 + x) / (x + -1.0)), -0.5)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * (Float64(Float64(1.0 + x) / Float64(x + -1.0)) ^ -0.5))
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 + x) / (x + -1.0)) ^ -0.5);
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[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

3. Simplified 29.9%

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

5. Taylor expanded in t around inf 37.9%

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

6. Taylor expanded in t around 0 38.0%

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

   1. clear-num 38.0%

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

   2. sub-neg 38.0%

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

   3. metadata-eval 38.0%

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

   4. sqrt-div 38.0%

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

   5. metadata-eval 38.0%

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

   6. +-commutative 38.0%

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

8. Applied egg-rr 38.0%

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

   1. pow1/2 38.0%

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

   2. pow-flip 38.0%

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

   3. +-commutative 38.0%

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

   4. metadata-eval 38.0%

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

10. Applied egg-rr 38.0%

    \[\leadsto \color{blue}{{\left(\frac{1 + x}{x + -1}\right)}^{-0.5}} \]
11. Add Preprocessing

Alternative 6: 77.4% accurate, 2.1× speedup

Math

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

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) (+ 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 * sqrt(((x - 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 * sqrt(((x - 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 * Math.sqrt(((x - 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 * math.sqrt(((x - 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 * sqrt(Float64(Float64(x - 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 * sqrt(((x - 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[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

3. Simplified 29.9%

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

5. Taylor expanded in t around inf 37.9%

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

6. Taylor expanded in t around 0 38.0%

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

Alternative 7: 76.7% accurate, 45.0× 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 35.7%

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

3. Simplified 29.9%

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

5. Taylor expanded in t around inf 37.9%

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

6. Taylor expanded in x around inf 37.5%

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

Alternative 8: 76.1% accurate, 225.0× 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 35.7%

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

3. Simplified 29.9%

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

5. Taylor expanded in t around inf 37.9%

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

6. Taylor expanded in x around inf 37.4%

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

Reproduce

herbie shell --seed 2024077 -o generate:simplify
(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)))))