Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 81.6%

Time: 18.7s

Alternatives: 10

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: 32.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2} + {l\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-223}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_2 + t\_2}{t\_m \cdot \left(\sqrt{2} \cdot x\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.7e-223)
      (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (fma x 0.5 -0.5))))
      (if (<= t_m 1.85e-58)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_2 t_2) (* t_m (* (sqrt 2.0) x))))
           (* t_m (sqrt 2.0)))))
        (sqrt (/ (+ x -1.0) (+ x 1.0))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0);
	double tmp;
	if (t_m <= 4.7e-223) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5)));
	} else if (t_m <= 1.85e-58) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_2 + t_2) / (t_m * (sqrt(2.0) * x)))) + (t_m * sqrt(2.0))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.7e-223)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5))));
	elseif (t_m <= 1.85e-58)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_2 + t_2) / Float64(t_m * Float64(sqrt(2.0) * x)))) + Float64(t_m * sqrt(2.0)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.7e-223], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e-58], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $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 < 4.70000000000000021e-223

   1. Initial program 31.6%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in l around inf 2.1%

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

      1. associate-/l* 2.1%

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

      2. associate--l+ 9.9%

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

      3. sub-neg 9.9%

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

      4. metadata-eval 9.9%

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

      5. +-commutative 9.9%

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

      6. sub-neg 9.9%

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

      7. metadata-eval 9.9%

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

      8. +-commutative 9.9%

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

   7. Simplified 9.9%

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

   8. Taylor expanded in x around 0 14.7%

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

   9. Taylor expanded in t around 0 14.8%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 14.7%

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

       2. *-commutative 14.7%

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

       3. fmm-def 14.7%

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

       4. metadata-eval 14.7%

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

       5. associate-*r* 16.0%

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

   11. Simplified 16.0%

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

   if 4.70000000000000021e-223 < t < 1.8500000000000001e-58

   1. Initial program 34.5%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in x around inf 72.2%

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

   if 1.8500000000000001e-58 < t

   1. Initial program 37.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in t around inf 92.9%

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

      1. associate-*l* 92.9%

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

      2. +-commutative 92.9%

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

      3. sub-neg 92.9%

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

      4. metadata-eval 92.9%

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

      5. +-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t around 0 94.4%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-223}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(\sqrt{2} \cdot x\right)} + t \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_2 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 7.2e-59)
      (*
       (sqrt 2.0)
       (/
        t_m
        (sqrt
         (+
          (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
          (/ (+ t_2 (pow l_m 2.0)) x)))))
      (sqrt (/ (+ x -1.0) (+ x 1.0)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 7.2e-59) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + ((t_2 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (t_m ** 2.0d0)
    if (t_m <= 7.2d-59) then
        tmp = sqrt(2.0d0) * (t_m / sqrt((((2.0d0 * ((t_m ** 2.0d0) / x)) + (t_2 + ((l_m ** 2.0d0) / x))) + ((t_2 + (l_m ** 2.0d0)) / x))))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * Math.pow(t_m, 2.0);
	double tmp;
	if (t_m <= 7.2e-59) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt((((2.0 * (Math.pow(t_m, 2.0) / x)) + (t_2 + (Math.pow(l_m, 2.0) / x))) + ((t_2 + Math.pow(l_m, 2.0)) / x))));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 * math.pow(t_m, 2.0)
	tmp = 0
	if t_m <= 7.2e-59:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt((((2.0 * (math.pow(t_m, 2.0) / x)) + (t_2 + (math.pow(l_m, 2.0) / x))) + ((t_2 + math.pow(l_m, 2.0)) / x))))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 7.2e-59)
		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_m ^ 2.0) / x))) + Float64(Float64(t_2 + (l_m ^ 2.0)) / x)))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 * (t_m ^ 2.0);
	tmp = 0.0;
	if (t_m <= 7.2e-59)
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * ((t_m ^ 2.0) / x)) + (t_2 + ((l_m ^ 2.0) / x))) + ((t_2 + (l_m ^ 2.0)) / x))));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := 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, 7.2e-59], 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$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[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 < 7.20000000000000001e-59

   1. Initial program 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in x around inf 57.0%

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

   if 7.20000000000000001e-59 < t

   1. Initial program 37.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in t around inf 92.9%

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

      1. associate-*l* 92.9%

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

      2. +-commutative 92.9%

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

      3. sub-neg 92.9%

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

      4. metadata-eval 92.9%

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

      5. +-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in t around 0 94.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-173}:\\ \;\;\;\;t\_m \cdot \left(\frac{\sqrt{2}}{l\_m} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-173)
    (* t_m (* (/ (sqrt 2.0) l_m) (sqrt (fma x 0.5 -0.5))))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 2.75e-173) {
		tmp = t_m * ((sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5)));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 2.75e-173)
		tmp = Float64(t_m * Float64(Float64(sqrt(2.0) / l_m) * sqrt(fma(x, 0.5, -0.5))));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-173], N[(t$95$m * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(x * 0.5 + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000011e-173

   1. Initial program 30.3%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in l around inf 2.0%

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

      1. associate-/l* 2.0%

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

      2. associate--l+ 11.0%

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

      3. sub-neg 11.0%

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

      4. metadata-eval 11.0%

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

      5. +-commutative 11.0%

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

      6. sub-neg 11.0%

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

      7. metadata-eval 11.0%

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

      8. +-commutative 11.0%

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

   7. Simplified 11.0%

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

   8. Taylor expanded in x around 0 16.6%

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

   9. Taylor expanded in t around 0 16.7%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\ell} \cdot \sqrt{0.5 \cdot x - 0.5}} \]
   10. Step-by-step derivation

       1. associate-*r/ 16.6%

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

       2. *-commutative 16.6%

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

       3. fmm-def 16.6%

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

       4. metadata-eval 16.6%

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

       5. associate-*r* 17.8%

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

   11. Simplified 17.8%

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

   if 2.75000000000000011e-173 < t

   1. Initial program 38.9%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in t around inf 87.6%

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

      1. associate-*l* 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in t around 0 88.8%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-173}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{\mathsf{fma}\left(x, 0.5, -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;t\_m \cdot \left(\frac{1}{l\_m} \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-181)
    (* t_m (* (/ 1.0 l_m) (sqrt x)))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = t_m * ((1.0 / l_m) * sqrt(x));
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.35d-181) then
        tmp = t_m * ((1.0d0 / l_m) * sqrt(x))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = t_m * ((1.0 / l_m) * Math.sqrt(x));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.35e-181:
		tmp = t_m * ((1.0 / l_m) * math.sqrt(x))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.35e-181)
		tmp = Float64(t_m * Float64(Float64(1.0 / l_m) * sqrt(x)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.35e-181)
		tmp = t_m * ((1.0 / l_m) * sqrt(x));
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-181], N[(t$95$m * N[(N[(1.0 / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e-181

   1. Initial program 30.3%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around inf 3.6%

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

      1. associate--l+ 13.0%

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

      2. sub-neg 13.0%

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

      3. metadata-eval 13.0%

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

      4. +-commutative 13.0%

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

      5. sub-neg 13.0%

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

      6. metadata-eval 13.0%

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

      7. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in x around inf 17.4%

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

      1. clear-num 17.3%

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

      2. un-div-inv 17.3%

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

      3. associate-*l* 17.3%

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

      4. sqrt-div 17.3%

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

      5. metadata-eval 17.3%

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

      6. un-div-inv 17.3%

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

   10. Applied egg-rr 17.3%

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

       1. associate-/r/ 17.4%

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

       2. *-lft-identity 17.4%

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

       3. times-frac 17.4%

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

       4. associate-/r/ 17.4%

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

       5. *-inverses 17.4%

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

   12. Simplified 17.4%

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

   if 1.35e-181 < t

   1. Initial program 38.9%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in t around inf 87.6%

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

      1. associate-*l* 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in t around 0 88.8%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{t\_m}{l\_m}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-181)
    (/ (/ t_m l_m) (pow x -0.5))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = (t_m / l_m) / pow(x, -0.5);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.35d-181) then
        tmp = (t_m / l_m) / (x ** (-0.5d0))
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = (t_m / l_m) / Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.35e-181:
		tmp = (t_m / l_m) / math.pow(x, -0.5)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.35e-181)
		tmp = Float64(Float64(t_m / l_m) / (x ^ -0.5));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.35e-181)
		tmp = (t_m / l_m) / (x ^ -0.5);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-181], N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e-181

   1. Initial program 30.3%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around inf 3.6%

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

      1. associate--l+ 13.0%

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

      2. sub-neg 13.0%

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

      3. metadata-eval 13.0%

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

      4. +-commutative 13.0%

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

      5. sub-neg 13.0%

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

      6. metadata-eval 13.0%

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

      7. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in x around inf 17.4%

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

      1. associate-/r* 16.2%

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

      2. associate-*r/ 16.2%

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

      3. *-commutative 16.2%

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

      4. inv-pow 16.2%

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

      5. sqrt-pow1 16.2%

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

      6. metadata-eval 16.2%

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

   10. Applied egg-rr 16.2%

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

       1. associate-*r/ 16.2%

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

       2. times-frac 16.2%

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

       3. *-inverses 16.2%

          \[\leadsto \frac{\color{blue}{1} \cdot \frac{t}{\ell}}{{x}^{-0.5}} \]

   12. Simplified 16.2%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 1.35e-181 < t

   1. Initial program 38.9%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in t around inf 87.6%

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

      1. associate-*l* 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in t around 0 88.8%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.66 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.66e-181)
    (* (sqrt x) (/ t_m l_m))
    (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.66e-181) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.66d-181) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.66e-181) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.66e-181:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.66e-181)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.66e-181)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.66e-181], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.66e-181

   1. Initial program 30.3%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around inf 3.6%

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

      1. associate--l+ 13.0%

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

      2. sub-neg 13.0%

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

      3. metadata-eval 13.0%

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

      4. +-commutative 13.0%

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

      5. sub-neg 13.0%

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

      6. metadata-eval 13.0%

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

      7. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in x around inf 17.4%

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

   9. Taylor expanded in t around 0 16.2%

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

   if 1.66e-181 < t

   1. Initial program 38.9%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in t around inf 87.6%

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

      1. associate-*l* 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in t around 0 88.8%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.66 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-181)
    (* (sqrt x) (/ t_m l_m))
    (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.35d-181) then
        tmp = sqrt(x) * (t_m / l_m)
    else
        tmp = 1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 1.35e-181) {
		tmp = Math.sqrt(x) * (t_m / l_m);
	} else {
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 1.35e-181:
		tmp = math.sqrt(x) * (t_m / l_m)
	else:
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 1.35e-181)
		tmp = Float64(sqrt(x) * Float64(t_m / l_m));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 1.35e-181)
		tmp = sqrt(x) * (t_m / l_m);
	else
		tmp = 1.0 + ((-1.0 - (-0.5 / x)) / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-181], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e-181

   1. Initial program 30.3%

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

   3. Simplified 30.2%

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

   5. Taylor expanded in l around inf 3.6%

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

      1. associate--l+ 13.0%

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

      2. sub-neg 13.0%

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

      3. metadata-eval 13.0%

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

      4. +-commutative 13.0%

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

      5. sub-neg 13.0%

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

      6. metadata-eval 13.0%

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

      7. +-commutative 13.0%

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

   7. Simplified 13.0%

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

   8. Taylor expanded in x around inf 17.4%

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

   9. Taylor expanded in t around 0 16.2%

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

   if 1.35e-181 < t

   1. Initial program 38.9%

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

   3. Simplified 30.5%

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

   5. Taylor expanded in t around inf 87.6%

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

      1. associate-*l* 87.6%

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

      2. +-commutative 87.6%

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

      3. sub-neg 87.6%

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

      4. metadata-eval 87.6%

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

      5. +-commutative 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in x around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. unsub-neg 0.0%

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

   10. Simplified 88.2%

       \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \frac{-0.5}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 25.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (+ 1.0 (/ (- -1.0 (/ -0.5 x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + (((-1.0d0) - ((-0.5d0) / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(Float64(-1.0 - Float64(-0.5 / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + ((-1.0 - (-0.5 / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(N[(-1.0 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 26.3%

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

5. Taylor expanded in t around inf 37.1%

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

   1. associate-*l* 37.1%

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

   2. +-commutative 37.1%

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

   3. sub-neg 37.1%

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

   4. metadata-eval 37.1%

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

   5. +-commutative 37.1%

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

7. Simplified 37.1%

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

8. Taylor expanded in x around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unsub-neg 0.0%

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

10. Simplified 37.3%

    \[\leadsto \color{blue}{1 - \frac{\frac{-0.5}{x} + 1}{x}} \]

11. Final simplification 37.3%

    \[\leadsto 1 + \frac{-1 - \frac{-0.5}{x}}{x} \]
12. Add Preprocessing

Alternative 9: 76.0% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 26.3%

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

5. Taylor expanded in t around inf 37.1%

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

   1. associate-*l* 37.1%

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

   2. +-commutative 37.1%

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

   3. sub-neg 37.1%

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

   4. metadata-eval 37.1%

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

   5. +-commutative 37.1%

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

7. Simplified 37.1%

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

8. Taylor expanded in x around inf 37.3%

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

9. Final simplification 37.3%

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

Alternative 10: 75.4% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 26.3%

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

5. Taylor expanded in t around inf 37.1%

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

   1. associate-*l* 37.1%

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

   2. +-commutative 37.1%

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

   3. sub-neg 37.1%

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

   4. metadata-eval 37.1%

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

   5. +-commutative 37.1%

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

7. Simplified 37.1%

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

8. Taylor expanded in x around inf 37.1%

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

Reproduce

herbie shell --seed 2024175 
(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)))))