Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 85.0%

Time: 8.6s

Alternatives: 11

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: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \frac{t\_2}{x}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(2, t\_2, \frac{\left(t\_3 + \left(t\_2 + t\_2\right)\right) + t\_3}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\frac{1 + x}{x - 1}} \cdot t\_4}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l)))
        (t_3 (/ t_2 x))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-172)
      (/ t_4 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l l) t_m) 2.0) t_4))
      (if (<= t_m 1.95e+51)
        (/
         t_4
         (sqrt
          (fma
           (* t_m t_m)
           2.0
           (/ (fma 2.0 t_2 (/ (+ (+ t_3 (+ t_2 t_2)) t_3) x)) x))))
        (/ t_4 (* (sqrt (/ (+ 1.0 x) (- x 1.0))) t_4)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = t_2 / x;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5.8e-172) {
		tmp = t_4 / fma((0.5 / (sqrt(2.0) * x)), (((l * l) / t_m) * 2.0), t_4);
	} else if (t_m <= 1.95e+51) {
		tmp = t_4 / sqrt(fma((t_m * t_m), 2.0, (fma(2.0, t_2, (((t_3 + (t_2 + t_2)) + t_3) / x)) / x)));
	} else {
		tmp = t_4 / (sqrt(((1.0 + x) / (x - 1.0))) * t_4);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(t_2 / x)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5.8e-172)
		tmp = Float64(t_4 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_4));
	elseif (t_m <= 1.95e+51)
		tmp = Float64(t_4 / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(fma(2.0, t_2, Float64(Float64(Float64(t_3 + Float64(t_2 + t_2)) + t_3) / x)) / x))));
	else
		tmp = Float64(t_4 / Float64(sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) * t_4));
	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[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-172], N[(t$95$4 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+51], N[(t$95$4 / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(2.0 * t$95$2 + N[(N[(N[(t$95$3 + N[(t$95$2 + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.79999999999999995e-172

   1. Initial program 24.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 14.9%

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

   if 5.79999999999999995e-172 < t < 1.94999999999999992e51

   1. Initial program 59.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 87.1%

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

   if 1.94999999999999992e51 < t

   1. Initial program 36.9%

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

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \left(\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) + \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}{x}\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \frac{t\_2}{x}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\mathsf{fma}\left(2, t\_2, t\_3\right) + t\_3}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\frac{1 + x}{x - 1}} \cdot t\_4}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l)))
        (t_3 (/ t_2 x))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 5.8e-172)
      (/ t_4 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l l) t_m) 2.0) t_4))
      (if (<= t_m 1.95e+51)
        (/ t_4 (sqrt (fma (* t_m t_m) 2.0 (/ (+ (fma 2.0 t_2 t_3) t_3) x))))
        (/ t_4 (* (sqrt (/ (+ 1.0 x) (- x 1.0))) t_4)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = t_2 / x;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 5.8e-172) {
		tmp = t_4 / fma((0.5 / (sqrt(2.0) * x)), (((l * l) / t_m) * 2.0), t_4);
	} else if (t_m <= 1.95e+51) {
		tmp = t_4 / sqrt(fma((t_m * t_m), 2.0, ((fma(2.0, t_2, t_3) + t_3) / x)));
	} else {
		tmp = t_4 / (sqrt(((1.0 + x) / (x - 1.0))) * t_4);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(t_2 / x)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 5.8e-172)
		tmp = Float64(t_4 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_4));
	elseif (t_m <= 1.95e+51)
		tmp = Float64(t_4 / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(Float64(fma(2.0, t_2, t_3) + t_3) / x))));
	else
		tmp = Float64(t_4 / Float64(sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) * t_4));
	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[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-172], N[(t$95$4 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+51], N[(t$95$4 / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(2.0 * t$95$2 + t$95$3), $MachinePrecision] + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.79999999999999995e-172

   1. Initial program 24.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 14.9%

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

   if 5.79999999999999995e-172 < t < 1.94999999999999992e51

   1. Initial program 59.1%

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

   3. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 86.7%

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

   if 1.94999999999999992e51 < t

   1. Initial program 36.9%

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

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t} \cdot 2, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right), \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(\sqrt{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, t\_3\right) + t\_3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{1 + x}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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_3 (/ (fma (* t_m t_m) 2.0 (* l l)) x)))
   (*
    t_s
    (if (<= t_m 5.8e-172)
      (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l l) t_m) 2.0) t_2))
      (if (<= t_m 1.95e+51)
        (/ t_2 (sqrt (+ (fma (* t_m t_m) 2.0 t_3) t_3)))
        (/ t_2 (* (sqrt (/ (+ 1.0 x) (- 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 t_3 = fma((t_m * t_m), 2.0, (l * l)) / x;
	double tmp;
	if (t_m <= 5.8e-172) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l * l) / t_m) * 2.0), t_2);
	} else if (t_m <= 1.95e+51) {
		tmp = t_2 / sqrt((fma((t_m * t_m), 2.0, t_3) + t_3));
	} else {
		tmp = t_2 / (sqrt(((1.0 + x) / (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)
	t_3 = Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) / x)
	tmp = 0.0
	if (t_m <= 5.8e-172)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_2));
	elseif (t_m <= 1.95e+51)
		tmp = Float64(t_2 / sqrt(Float64(fma(Float64(t_m * t_m), 2.0, t_3) + t_3)));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(1.0 + x) / 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]}, Block[{t$95$3 = N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-172], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.95e+51], N[(t$95$2 / N[Sqrt[N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + t$95$3), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.79999999999999995e-172

   1. Initial program 24.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 14.9%

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

   if 5.79999999999999995e-172 < t < 1.94999999999999992e51

   1. Initial program 59.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   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. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-+.f64 N/A

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

   5. Applied rewrites 85.9%

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

   if 1.94999999999999992e51 < t

   1. Initial program 36.9%

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

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 93.6

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

   5. Applied rewrites 93.6%

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

Alternative 4: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{0.5}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t\_m} \cdot 2, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\frac{1 + x}{x - 1}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 4e-54)
      (/ t_2 (fma (/ 0.5 (* (sqrt 2.0) x)) (* (/ (* l l) t_m) 2.0) t_2))
      (/ t_2 (* (sqrt (/ (+ 1.0 x) (- 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 <= 4e-54) {
		tmp = t_2 / fma((0.5 / (sqrt(2.0) * x)), (((l * l) / t_m) * 2.0), t_2);
	} else {
		tmp = t_2 / (sqrt(((1.0 + x) / (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 <= 4e-54)
		tmp = Float64(t_2 / fma(Float64(0.5 / Float64(sqrt(2.0) * x)), Float64(Float64(Float64(l * l) / t_m) * 2.0), t_2));
	else
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(1.0 + x) / 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, 4e-54], N[(t$95$2 / N[(N[(0.5 / N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000001e-54

   1. Initial program 27.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 19.8%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 19.6%

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

   if 4.0000000000000001e-54 < t

   1. Initial program 44.2%

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

   3. Taylor expanded in l around 0

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

      1. *-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. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 87.9

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

   5. Applied rewrites 87.9%

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

Alternative 5: 77.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \frac{t\_2}{\sqrt{\frac{1 + x}{x - 1}} \cdot t\_2} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (* t_s (/ t_2 (* (sqrt (/ (+ 1.0 x) (- 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;
	return t_s * (t_2 / (sqrt(((1.0 + x) / (x - 1.0))) * t_2));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: t_2
    t_2 = sqrt(2.0d0) * t_m
    code = t_s * (t_2 / (sqrt(((1.0d0 + x) / (x - 1.0d0))) * t_2))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double t_2 = Math.sqrt(2.0) * t_m;
	return t_s * (t_2 / (Math.sqrt(((1.0 + x) / (x - 1.0))) * t_2));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = math.sqrt(2.0) * t_m
	return t_s * (t_2 / (math.sqrt(((1.0 + x) / (x - 1.0))) * t_2))

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)
	return Float64(t_s * Float64(t_2 / Float64(sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0))) * t_2)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = t_s * (t_2 / (sqrt(((1.0 + x) / (x - 1.0))) * t_2));
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 * N[(t$95$2 / N[(N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.4%

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

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 34.5

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

5. Applied rewrites 34.5%

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

Alternative 6: 77.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{t\_m \cdot \sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m} \end{array} \]

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 (/ (* t_m (sqrt 2.0)) (* (sqrt (/ (fma x 2.0 2.0) (- x 1.0))) t_m))))

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 * ((t_m * sqrt(2.0)) / (sqrt((fma(x, 2.0, 2.0) / (x - 1.0))) * t_m));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(t_m * sqrt(2.0)) / Float64(sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x - 1.0))) * t_m)))
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 34.5

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

5. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 16.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 16.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 16.6

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

9. Applied rewrites 34.5%

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

Alternative 7: 77.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{t\_m}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m} \cdot \sqrt{2}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* (/ t_m (* (sqrt (/ (fma x 2.0 2.0) (- 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) {
	return t_s * ((t_m / (sqrt((fma(x, 2.0, 2.0) / (x - 1.0))) * t_m)) * sqrt(2.0));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(t_m / Float64(sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x - 1.0))) * t_m)) * sqrt(2.0)))
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 34.5

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

5. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 16.6%

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

9. Applied rewrites 34.4%

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

Alternative 8: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(t\_m \cdot \frac{\sqrt{2}}{\sqrt{\frac{\mathsf{fma}\left(x, 2, 2\right)}{x - 1}} \cdot t\_m}\right) \end{array} \]

FPCore

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

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 * (t_m * (sqrt(2.0) / (sqrt((fma(x, 2.0, 2.0) / (x - 1.0))) * t_m)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(t_m * Float64(sqrt(2.0) / Float64(sqrt(Float64(fma(x, 2.0, 2.0) / Float64(x - 1.0))) * t_m))))
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 34.5

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

5. Applied rewrites 34.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 16.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 16.6

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

9. Applied rewrites 34.3%

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

Alternative 9: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\sqrt{\frac{x - 1}{1 + x}} \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* (* (sqrt (/ (- x 1.0) (+ 1.0 x))) (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) {
	return t_s * ((sqrt(((x - 1.0) / (1.0 + x))) * sqrt(0.5)) * sqrt(2.0));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * ((sqrt(((x - 1.0d0) / (1.0d0 + x))) * sqrt(0.5d0)) * sqrt(2.0d0))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * ((Math.sqrt(((x - 1.0) / (1.0 + x))) * Math.sqrt(0.5)) * Math.sqrt(2.0));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * ((math.sqrt(((x - 1.0) / (1.0 + x))) * math.sqrt(0.5)) * math.sqrt(2.0))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(Float64(sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))) * sqrt(0.5)) * sqrt(2.0)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * ((sqrt(((x - 1.0) / (1.0 + x))) * sqrt(0.5)) * sqrt(2.0));
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 39.3%

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

5. Taylor expanded in l around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-sqrt.f64 34.0

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

7. Applied rewrites 34.0%

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

Alternative 10: 76.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\sqrt{\frac{x - 1}{1 + x}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (* t_s (* (sqrt (/ (- x 1.0) (+ 1.0 x))) (* (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) {
	return t_s * (sqrt(((x - 1.0) / (1.0 + x))) * (sqrt(0.5) * sqrt(2.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * (sqrt(((x - 1.0d0) / (1.0d0 + x))) * (sqrt(0.5d0) * sqrt(2.0d0)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	return t_s * (Math.sqrt(((x - 1.0) / (1.0 + x))) * (Math.sqrt(0.5) * Math.sqrt(2.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	return t_s * (math.sqrt(((x - 1.0) / (1.0 + x))) * (math.sqrt(0.5) * math.sqrt(2.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	return Float64(t_s * Float64(sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))) * Float64(sqrt(0.5) * sqrt(2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l, t_m)
	tmp = t_s * (sqrt(((x - 1.0) / (1.0 + x))) * (sqrt(0.5) * sqrt(2.0)));
end

Wolfram

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

Derivation

1. Initial program 32.4%

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

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 34.0

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

5. Applied rewrites 34.0%

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

Alternative 11: 76.1% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.4%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
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 33.3

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

5. Applied rewrites 33.3%

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

7. Applied rewrites 33.8%

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

Reproduce

herbie shell --seed 2024332 
(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)))))