Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 84.1%

Time: 10.0s

Alternatives: 10

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.2% 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: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{-8}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{x}, 2, \left(\frac{4}{x} + 2\right) \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 6.2e-166)
      (/ t_2 (fma (/ (* (* l l) 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
      (if (<= t_m 3e-8)
        (*
         t_m
         (sqrt
          (/ 2.0 (fma (/ (* l l) x) 2.0 (* (+ (/ 4.0 x) 2.0) (* t_m t_m))))))
        (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2)))))))

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 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 6.2e-166) {
		tmp = t_2 / fma((((l * l) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 3e-8) {
		tmp = t_m * sqrt((2.0 / fma(((l * l) / x), 2.0, (((4.0 / x) + 2.0) * (t_m * t_m)))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	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(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 6.2e-166)
		tmp = Float64(t_2 / fma(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 3e-8)
		tmp = Float64(t_m * sqrt(Float64(2.0 / fma(Float64(Float64(l * l) / x), 2.0, Float64(Float64(Float64(4.0 / x) + 2.0) * Float64(t_m * t_m))))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.2e-166], N[(t$95$2 / N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e-8], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999968e-166

   1. Initial program 33.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 14.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 14.5%

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

   if 6.19999999999999968e-166 < t < 2.99999999999999973e-8

   1. Initial program 45.2%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-add N/A

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

      10. associate-*r/ N/A

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

      11. remove-double-neg N/A

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

   5. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

   7. Applied rewrites 81.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 81.9%

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

   if 2.99999999999999973e-8 < t

   1. Initial program 35.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 2: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{x}^{-1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2 \cdot \frac{x - -1}{x - 1}} \cdot t\_m} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-216)
    (/ (* (sqrt 2.0) t_m) (* (* (sqrt 2.0) l) (sqrt (pow x -1.0))))
    (* (/ t_m (* (sqrt (* 2.0 (/ (- x -1.0) (- x 1.0)))) t_m)) (sqrt 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (sqrt(2.0) * t_m) / ((sqrt(2.0) * l) * sqrt(pow(x, -1.0)));
	} else {
		tmp = (t_m / (sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * sqrt(2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = (sqrt(2.0d0) * t_m) / ((sqrt(2.0d0) * l) * sqrt((x ** (-1.0d0))))
    else
        tmp = (t_m / (sqrt((2.0d0 * ((x - (-1.0d0)) / (x - 1.0d0)))) * t_m)) * sqrt(2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (Math.sqrt(2.0) * t_m) / ((Math.sqrt(2.0) * l) * Math.sqrt(Math.pow(x, -1.0)));
	} else {
		tmp = (t_m / (Math.sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * Math.sqrt(2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = (math.sqrt(2.0) * t_m) / ((math.sqrt(2.0) * l) * math.sqrt(math.pow(x, -1.0)))
	else:
		tmp = (t_m / (math.sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * math.sqrt(2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(sqrt(2.0) * l) * sqrt((x ^ -1.0))));
	else
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(2.0 * Float64(Float64(x - -1.0) / Float64(x - 1.0)))) * t_m)) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = (sqrt(2.0) * t_m) / ((sqrt(2.0) * l) * sqrt((x ^ -1.0)));
	else
		tmp = (t_m / (sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * sqrt(2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / N[(N[Sqrt[N[(2.0 * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. div-add-rev N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.7%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 82.9%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{{x}^{-1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \frac{x - -1}{x - 1}} \cdot t} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{\ell}\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{-8}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{x}, 2, \left(\frac{4}{x} + 2\right) \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{x - -1}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 7.2e-165)
      (/ (* (sqrt x) t_m) l)
      (if (<= t_m 3e-8)
        (*
         t_m
         (sqrt
          (/ 2.0 (fma (/ (* l l) x) 2.0 (* (+ (/ 4.0 x) 2.0) (* t_m t_m))))))
        (/ t_2 (* (sqrt (/ (- x -1.0) (- x 1.0))) t_2)))))))

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 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = (sqrt(x) * t_m) / l;
	} else if (t_m <= 3e-8) {
		tmp = t_m * sqrt((2.0 / fma(((l * l) / x), 2.0, (((4.0 / x) + 2.0) * (t_m * t_m)))));
	} else {
		tmp = t_2 / (sqrt(((x - -1.0) / (x - 1.0))) * t_2);
	}
	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(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(Float64(sqrt(x) * t_m) / l);
	elseif (t_m <= 3e-8)
		tmp = Float64(t_m * sqrt(Float64(2.0 / fma(Float64(Float64(l * l) / x), 2.0, Float64(Float64(Float64(4.0 / x) + 2.0) * Float64(t_m * t_m))))));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x - -1.0) / Float64(x - 1.0))) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$m, 3e-8], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 33.4%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.7%

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

   if 7.19999999999999969e-165 < t < 2.99999999999999973e-8

   1. Initial program 46.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-add N/A

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

      10. associate-*r/ N/A

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

      11. remove-double-neg N/A

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

   5. Applied rewrites 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 84.0%

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

   if 2.99999999999999973e-8 < t

   1. Initial program 35.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 4: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{\ell}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+34}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{x}, 2, \left(\frac{4}{x} + 2\right) \cdot \left(t\_m \cdot t\_m\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2 \cdot \frac{x - -1}{x - 1}} \cdot t\_m} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-165)
    (/ (* (sqrt x) t_m) l)
    (if (<= t_m 2e+34)
      (*
       t_m
       (sqrt
        (/ 2.0 (fma (/ (* l l) x) 2.0 (* (+ (/ 4.0 x) 2.0) (* t_m t_m))))))
      (*
       (/ t_m (* (sqrt (* 2.0 (/ (- x -1.0) (- x 1.0)))) t_m))
       (sqrt 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 7.2e-165) {
		tmp = (sqrt(x) * t_m) / l;
	} else if (t_m <= 2e+34) {
		tmp = t_m * sqrt((2.0 / fma(((l * l) / x), 2.0, (((4.0 / x) + 2.0) * (t_m * t_m)))));
	} else {
		tmp = (t_m / (sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * sqrt(2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 7.2e-165)
		tmp = Float64(Float64(sqrt(x) * t_m) / l);
	elseif (t_m <= 2e+34)
		tmp = Float64(t_m * sqrt(Float64(2.0 / fma(Float64(Float64(l * l) / x), 2.0, Float64(Float64(Float64(4.0 / x) + 2.0) * Float64(t_m * t_m))))));
	else
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(2.0 * Float64(Float64(x - -1.0) / Float64(x - 1.0)))) * t_m)) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-165], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[t$95$m, 2e+34], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] * 2.0 + N[(N[(N[(4.0 / x), $MachinePrecision] + 2.0), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / N[(N[Sqrt[N[(2.0 * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.19999999999999969e-165

   1. Initial program 33.4%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.7%

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

   if 7.19999999999999969e-165 < t < 1.99999999999999989e34

   1. Initial program 54.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-add N/A

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

      10. associate-*r/ N/A

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

      11. remove-double-neg N/A

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

   5. Applied rewrites 86.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

   7. Applied rewrites 86.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.9%

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

   if 1.99999999999999989e34 < t

   1. Initial program 29.5%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 95.7

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

   5. Applied rewrites 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 95.3%

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

Alternative 5: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\sqrt{2 \cdot \frac{x - -1}{x - 1}} \cdot t\_m} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-216)
    (/ (* (sqrt x) t_m) l)
    (* (/ t_m (* (sqrt (* 2.0 (/ (- x -1.0) (- x 1.0)))) t_m)) (sqrt 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (sqrt(x) * t_m) / l;
	} else {
		tmp = (t_m / (sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * sqrt(2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = (sqrt(x) * t_m) / l
    else
        tmp = (t_m / (sqrt((2.0d0 * ((x - (-1.0d0)) / (x - 1.0d0)))) * t_m)) * sqrt(2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (Math.sqrt(x) * t_m) / l;
	} else {
		tmp = (t_m / (Math.sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * Math.sqrt(2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = (math.sqrt(x) * t_m) / l
	else:
		tmp = (t_m / (math.sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * math.sqrt(2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(Float64(sqrt(x) * t_m) / l);
	else
		tmp = Float64(Float64(t_m / Float64(sqrt(Float64(2.0 * Float64(Float64(x - -1.0) / Float64(x - 1.0)))) * t_m)) * sqrt(2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = (sqrt(x) * t_m) / l;
	else
		tmp = (t_m / (sqrt((2.0 * ((x - -1.0) / (x - 1.0)))) * t_m)) * sqrt(2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision], N[(N[(t$95$m / N[(N[Sqrt[N[(2.0 * N[(N[(x - -1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.6%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 83.2

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

   5. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 82.9%

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

Alternative 6: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - 1}{x - -1}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.45e-216)
    (/ (* (sqrt x) t_m) l)
    (* (sqrt (/ (- x 1.0) (- x -1.0))) (* (sqrt 0.5) (sqrt 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (sqrt(x) * t_m) / l;
	} else {
		tmp = sqrt(((x - 1.0) / (x - -1.0))) * (sqrt(0.5) * sqrt(2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = (sqrt(x) * t_m) / l
    else
        tmp = sqrt(((x - 1.0d0) / (x - (-1.0d0)))) * (sqrt(0.5d0) * sqrt(2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (Math.sqrt(x) * t_m) / l;
	} else {
		tmp = Math.sqrt(((x - 1.0) / (x - -1.0))) * (Math.sqrt(0.5) * Math.sqrt(2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = (math.sqrt(x) * t_m) / l
	else:
		tmp = math.sqrt(((x - 1.0) / (x - -1.0))) * (math.sqrt(0.5) * math.sqrt(2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(Float64(sqrt(x) * t_m) / l);
	else
		tmp = Float64(sqrt(Float64(Float64(x - 1.0) / Float64(x - -1.0))) * Float64(sqrt(0.5) * sqrt(2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = (sqrt(x) * t_m) / l;
	else
		tmp = sqrt(((x - 1.0) / (x - -1.0))) * (sqrt(0.5) * sqrt(2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision], N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.6%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 81.9

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

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{x - -1}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (if (<= t_m 3.45e-216) (/ (* (sqrt x) t_m) l) 1.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (sqrt(x) * t_m) / l;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = (sqrt(x) * t_m) / l
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (Math.sqrt(x) * t_m) / l;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = (math.sqrt(x) * t_m) / l
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(Float64(sqrt(x) * t_m) / l);
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = (sqrt(x) * t_m) / l;
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.6%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 82.0%

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

Alternative 8: 77.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{x}}{\ell} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (if (<= t_m 3.45e-216) (* (/ (sqrt x) l) t_m) 1.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (sqrt(x) / l) * t_m;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = (sqrt(x) / l) * t_m
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = (Math.sqrt(x) / l) * t_m;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = (math.sqrt(x) / l) * t_m
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(Float64(sqrt(x) / l) * t_m);
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = (sqrt(x) / l) * t_m;
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.6%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 82.0%

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

Alternative 9: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.45 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t\_m}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (if (<= t_m 3.45e-216) (* (sqrt x) (/ t_m l)) 1.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.45d-216) then
        tmp = sqrt(x) * (t_m / l)
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 3.45e-216) {
		tmp = Math.sqrt(x) * (t_m / l);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 3.45e-216:
		tmp = math.sqrt(x) * (t_m / l)
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 3.45e-216)
		tmp = Float64(sqrt(x) * Float64(t_m / l));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 3.45e-216)
		tmp = sqrt(x) * (t_m / l);
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.45e-216], N[(N[Sqrt[x], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4500000000000001e-216

   1. Initial program 34.7%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 16.6%

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

   10. Applied rewrites 16.1%

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

   if 3.4500000000000001e-216 < t

   1. Initial program 37.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 82.0%

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

Alternative 10: 75.3% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.9%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-sqrt.f64 37.8

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

5. Applied rewrites 37.8%

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

7. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2024305 
(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)))))