Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 79.9%

Time: 19.0s

Alternatives: 7

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% 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: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m \cdot {\left(\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}\right)}^{-0.5}}{l\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-190)
    (*
     (sqrt 2.0)
     (/ (* t_m (pow (+ (/ 1.0 (+ x -1.0)) (/ (+ 1.0 (/ 1.0 x)) x)) -0.5)) l_m))
    (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-190) {
		tmp = sqrt(2.0) * ((t_m * pow(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)), -0.5)) / l_m);
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.6d-190) then
        tmp = sqrt(2.0d0) * ((t_m * (((1.0d0 / (x + (-1.0d0))) + ((1.0d0 + (1.0d0 / x)) / x)) ** (-0.5d0))) / l_m)
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.6e-190) {
		tmp = Math.sqrt(2.0) * ((t_m * Math.pow(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)), -0.5)) / l_m);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.6e-190:
		tmp = math.sqrt(2.0) * ((t_m * math.pow(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)), -0.5)) / l_m)
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.6e-190)
		tmp = Float64(sqrt(2.0) * Float64(Float64(t_m * (Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 + Float64(1.0 / x)) / x)) ^ -0.5)) / l_m));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.6e-190)
		tmp = sqrt(2.0) * ((t_m * (((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)) ^ -0.5)) / l_m);
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-190], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.60000000000000007e-190

   1. Initial program 30.9%

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

   3. Simplified 26.6%

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

      1. add-sqr-sqrt 14.8%

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

      2. pow2 14.8%

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

   6. Applied egg-rr 14.8%

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

   7. Taylor expanded in l around inf 3.1%

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

      1. associate--l+ 10.0%

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

      2. sub-neg 10.0%

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

      3. metadata-eval 10.0%

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

      4. +-commutative 10.0%

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

      5. sub-neg 10.0%

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

      6. metadata-eval 10.0%

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

      7. +-commutative 10.0%

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

   9. Simplified 10.0%

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

   10. Taylor expanded in x around inf 16.0%

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

       1. associate-*l/ 18.4%

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

       2. inv-pow 18.4%

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

       3. sqrt-pow1 18.4%

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

       4. +-commutative 18.4%

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

       5. metadata-eval 18.4%

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

   12. Applied egg-rr 18.4%

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

   if 3.60000000000000007e-190 < t

   1. Initial program 38.8%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in t around inf 86.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t \cdot {\left(\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}\right)}^{-0.5}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_m \cdot \frac{\sqrt{x \cdot 0.5}}{l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-189)
    (* (sqrt 2.0) (* t_m (/ (sqrt (* x 0.5)) l_m)))
    (sqrt (/ (+ x -1.0) (+ 1.0 x))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.5e-189) {
		tmp = sqrt(2.0) * (t_m * (sqrt((x * 0.5)) / l_m));
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 3.5d-189) then
        tmp = sqrt(2.0d0) * (t_m * (sqrt((x * 0.5d0)) / l_m))
    else
        tmp = sqrt(((x + (-1.0d0)) / (1.0d0 + x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (t_m <= 3.5e-189) {
		tmp = Math.sqrt(2.0) * (t_m * (Math.sqrt((x * 0.5)) / l_m));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (1.0 + x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if t_m <= 3.5e-189:
		tmp = math.sqrt(2.0) * (t_m * (math.sqrt((x * 0.5)) / l_m))
	else:
		tmp = math.sqrt(((x + -1.0) / (1.0 + x)))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (t_m <= 3.5e-189)
		tmp = Float64(sqrt(2.0) * Float64(t_m * Float64(sqrt(Float64(x * 0.5)) / l_m)));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (t_m <= 3.5e-189)
		tmp = sqrt(2.0) * (t_m * (sqrt((x * 0.5)) / l_m));
	else
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-189], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sqrt[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000001e-189

   1. Initial program 30.9%

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

   3. Simplified 26.6%

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

      1. add-sqr-sqrt 14.8%

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

      2. pow2 14.8%

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

   6. Applied egg-rr 14.8%

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

   7. Taylor expanded in l around inf 3.1%

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

      1. associate--l+ 10.0%

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

      2. sub-neg 10.0%

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

      3. metadata-eval 10.0%

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

      4. +-commutative 10.0%

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

      5. sub-neg 10.0%

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

      6. metadata-eval 10.0%

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

      7. +-commutative 10.0%

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

   9. Simplified 10.0%

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

   10. Taylor expanded in x around inf 15.6%

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

       1. *-commutative 15.6%

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

   12. Simplified 15.6%

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

       1. associate-*l/ 18.0%

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

   14. Applied egg-rr 18.0%

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

       1. associate-/l* 18.1%

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

   16. Simplified 18.1%

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

   if 3.5000000000000001e-189 < t

   1. Initial program 38.8%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in t around inf 86.5%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{2} \cdot \left(t \cdot \frac{\sqrt{x \cdot 0.5}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in t around inf 40.9%

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

6. Final simplification 40.9%

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

Alternative 4: 76.0% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in t around inf 40.9%

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

6. Taylor expanded in x around inf 40.9%

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

7. Taylor expanded in x around -inf 40.7%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.5 \cdot \frac{1}{x}}{x}}{x}} \]
8. Step-by-step derivation

   1. mul-1-neg 40.7%

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

   2. unsub-neg 40.7%

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

   3. mul-1-neg 40.7%

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

   4. unsub-neg 40.7%

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

   5. sub-neg 40.7%

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

   6. associate-*r/ 40.7%

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

   7. metadata-eval 40.7%

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

   8. distribute-neg-frac 40.7%

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

   9. metadata-eval 40.7%

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

9. Simplified 40.7%

   \[\leadsto \color{blue}{1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}} \]
10. Add Preprocessing

Alternative 5: 75.9% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in t around inf 40.9%

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

6. Taylor expanded in x around inf 40.9%

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

7. Taylor expanded in x around -inf 40.7%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 - 0.5 \cdot \frac{1}{x}}{x}} \]
8. Step-by-step derivation

   1. mul-1-neg 40.7%

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

   2. unsub-neg 40.7%

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

   3. sub-neg 40.7%

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

   4. associate-*r/ 40.7%

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

   5. metadata-eval 40.7%

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

   6. distribute-neg-frac 40.7%

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

   7. metadata-eval 40.7%

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

9. Simplified 40.7%

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

10. Final simplification 40.7%

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

Alternative 6: 75.6% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in l around 0 40.9%

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

   1. associate-*l/ 40.9%

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

   2. +-commutative 40.9%

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

   3. sub-neg 40.9%

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

   4. metadata-eval 40.9%

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

   5. *-lft-identity 40.9%

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

   6. +-commutative 40.9%

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

7. Simplified 40.9%

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

8. Taylor expanded in x around inf 40.7%

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

9. Final simplification 40.7%

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

Alternative 7: 75.0% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

4. Applied egg-rr 63.7%

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

5. Taylor expanded in t around inf 40.9%

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

6. Taylor expanded in x around inf 40.9%

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

7. Taylor expanded in x around inf 40.4%

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

Reproduce

herbie shell --seed 2024148 
(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)))))