Toniolo and Linder, Equation (7)

Percentage Accurate: 33.9% → 98.8%

Time: 19.8s

Alternatives: 8

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3100000:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, t\_2\right), \sqrt{2} \cdot \frac{t\_m}{\sqrt{x}}\right), \frac{\mathsf{hypot}\left(\ell, t\_2\right)}{\sqrt{x}}\right)}\\ \end{array} \end{array} \end{array} \]

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 (<= x 3100000.0)
      (sqrt (/ (+ x -1.0) (+ x 1.0)))
      (/
       t_2
       (hypot
        (hypot (hypot (/ l (sqrt x)) t_2) (* (sqrt 2.0) (/ t_m (sqrt x))))
        (/ (hypot l t_2) (sqrt x))))))))

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 (x <= 3100000.0) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_2 / hypot(hypot(hypot((l / sqrt(x)), t_2), (sqrt(2.0) * (t_m / sqrt(x)))), (hypot(l, t_2) / sqrt(x)));
	}
	return t_s * tmp;
}

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;
	double tmp;
	if (x <= 3100000.0) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = t_2 / Math.hypot(Math.hypot(Math.hypot((l / Math.sqrt(x)), t_2), (Math.sqrt(2.0) * (t_m / Math.sqrt(x)))), (Math.hypot(l, t_2) / Math.sqrt(x)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	t_2 = math.sqrt(2.0) * t_m
	tmp = 0
	if x <= 3100000.0:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = t_2 / math.hypot(math.hypot(math.hypot((l / math.sqrt(x)), t_2), (math.sqrt(2.0) * (t_m / math.sqrt(x)))), (math.hypot(l, t_2) / math.sqrt(x)))
	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 (x <= 3100000.0)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(t_2 / hypot(hypot(hypot(Float64(l / sqrt(x)), t_2), Float64(sqrt(2.0) * Float64(t_m / sqrt(x)))), Float64(hypot(l, t_2) / sqrt(x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	t_2 = sqrt(2.0) * t_m;
	tmp = 0.0;
	if (x <= 3100000.0)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = t_2 / hypot(hypot(hypot((l / sqrt(x)), t_2), (sqrt(2.0) * (t_m / sqrt(x)))), (hypot(l, t_2) / sqrt(x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[x, 3100000.0], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$2 / N[Sqrt[N[Sqrt[N[Sqrt[N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision] ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2 + N[(N[Sqrt[l ^ 2 + t$95$2 ^ 2], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e6

   1. Initial program 43.6%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in t around inf 46.2%

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

      1. associate-*l* 46.2%

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

      2. +-commutative 46.2%

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

      3. sub-neg 46.2%

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

      4. metadata-eval 46.2%

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

      5. +-commutative 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in t around 0 46.3%

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

   if 3.1e6 < x

   1. Initial program 27.3%

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

   3. Taylor expanded in x around inf 50.1%

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

   5. Applied egg-rr 98.8%

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

      1. *-lft-identity 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{hypot}\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \sqrt{2} \cdot t\right), \sqrt{2} \cdot \frac{t}{\sqrt{x}}\right), \frac{\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right)}{\sqrt{x}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3100000:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \sqrt{2} \cdot t\right), \sqrt{2} \cdot \frac{t}{\sqrt{x}}\right), \frac{\mathsf{hypot}\left(\ell, \sqrt{2} \cdot t\right)}{\sqrt{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 1.45e+174)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (*
     (sqrt 2.0)
     (/ t_m (* l (sqrt (+ (/ 1.0 (+ x -1.0)) (/ (+ 1.0 (/ 1.0 x)) x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.45e+174) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 1.45d+174) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 + (1.0d0 / x)) / x)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 1.45e+174) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 1.45e+174:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 1.45e+174)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 + Float64(1.0 / x)) / x))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 1.45e+174)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + ((1.0 + (1.0 / x)) / x)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 1.45e+174], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.45e174

   1. Initial program 36.6%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in t around inf 38.6%

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

      1. associate-*l* 38.6%

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

      2. +-commutative 38.6%

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

      3. sub-neg 38.6%

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

      4. metadata-eval 38.6%

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

      5. +-commutative 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in t around 0 38.7%

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

   if 1.45e174 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 9.9%

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

      1. associate--l+ 42.1%

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

      2. sub-neg 42.1%

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

      3. metadata-eval 42.1%

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

      4. +-commutative 42.1%

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

      5. sub-neg 42.1%

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

      6. sub-neg 42.1%

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

      7. metadata-eval 42.1%

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

      8. +-commutative 42.1%

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

      9. metadata-eval 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in x around inf 81.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x + -1} + \frac{1 + \frac{1}{x}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\ell \cdot \sqrt{\frac{1}{x + -1} + \frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= l 3.5e+173)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (* (sqrt 2.0) (/ t_m (* l (sqrt (+ (/ 1.0 (+ x -1.0)) (/ 1.0 x)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 3.5e+173) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + (1.0 / x)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l <= 3.5d+173) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = sqrt(2.0d0) * (t_m / (l * sqrt(((1.0d0 / (x + (-1.0d0))) + (1.0d0 / x)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (l <= 3.5e+173) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = Math.sqrt(2.0) * (t_m / (l * Math.sqrt(((1.0 / (x + -1.0)) + (1.0 / x)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if l <= 3.5e+173:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = math.sqrt(2.0) * (t_m / (l * math.sqrt(((1.0 / (x + -1.0)) + (1.0 / x)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (l <= 3.5e+173)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(l * sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(1.0 / x))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (l <= 3.5e+173)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = sqrt(2.0) * (t_m / (l * sqrt(((1.0 / (x + -1.0)) + (1.0 / x)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[l, 3.5e+173], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.4999999999999999e173

   1. Initial program 36.6%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in t around inf 38.6%

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

      1. associate-*l* 38.6%

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

      2. +-commutative 38.6%

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

      3. sub-neg 38.6%

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

      4. metadata-eval 38.6%

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

      5. +-commutative 38.6%

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

   7. Simplified 38.6%

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

   8. Taylor expanded in t around 0 38.7%

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

   if 3.4999999999999999e173 < l

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in l around inf 9.9%

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

      1. associate--l+ 42.1%

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

      2. sub-neg 42.1%

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

      3. metadata-eval 42.1%

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

      4. +-commutative 42.1%

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

      5. sub-neg 42.1%

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

      6. sub-neg 42.1%

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

      7. metadata-eval 42.1%

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

      8. +-commutative 42.1%

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

      9. metadata-eval 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in x around inf 79.3%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{+173}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{1}{x + -1} + \frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 2.1× speedup

Math

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

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 (sqrt (/ (+ x -1.0) (+ x 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 * sqrt(((x + -1.0) / (x + 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 * sqrt(((x + (-1.0d0)) / (x + 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 * Math.sqrt(((x + -1.0) / (x + 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 * math.sqrt(((x + -1.0) / (x + 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 * sqrt(Float64(Float64(x + -1.0) / Float64(x + 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 * sqrt(((x + -1.0) / (x + 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 * N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.6%

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

3. Simplified 28.7%

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

5. Taylor expanded in t around inf 36.5%

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

   1. associate-*l* 36.5%

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

   2. +-commutative 36.5%

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

   3. sub-neg 36.5%

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

   4. metadata-eval 36.5%

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

   5. +-commutative 36.5%

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

7. Simplified 36.5%

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

8. Taylor expanded in t around 0 36.6%

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

9. Final simplification 36.6%

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

Alternative 5: 76.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 28.7%

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

5. Taylor expanded in t around inf 36.5%

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

   1. associate-*l* 36.5%

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

   2. +-commutative 36.5%

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

   3. sub-neg 36.5%

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

   4. metadata-eval 36.5%

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

   5. +-commutative 36.5%

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

7. Simplified 36.5%

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

8. Taylor expanded in t around 0 36.6%

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

   1. add-sqr-sqrt 36.5%

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

   2. pow2 36.5%

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

   3. pow1/2 36.5%

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

   4. sqrt-pow1 36.5%

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

   5. sub-neg 36.5%

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

   6. metadata-eval 36.5%

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

   7. +-commutative 36.5%

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

   8. metadata-eval 36.5%

      \[\leadsto {\left({\left(\frac{x + -1}{x + 1}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

10. Applied egg-rr 36.5%

    \[\leadsto \color{blue}{{\left({\left(\frac{x + -1}{x + 1}\right)}^{0.25}\right)}^{2}} \]

11. Taylor expanded in x around -inf 36.6%

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

    1. mul-1-neg 36.6%

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

    2. unsub-neg 36.6%

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

    3. mul-1-neg 36.6%

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

    4. unsub-neg 36.6%

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

    5. sub-neg 36.6%

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

    6. associate-*r/ 36.6%

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

    7. metadata-eval 36.6%

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

    8. distribute-neg-frac 36.6%

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

    9. metadata-eval 36.6%

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

13. Simplified 36.6%

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

14. Final simplification 36.6%

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

Alternative 6: 76.1% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 28.7%

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

5. Taylor expanded in t around inf 36.5%

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

   1. rem-exp-log 32.5%

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

   2. *-un-lft-identity 32.5%

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

   3. exp-prod 32.4%

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

   4. exp-1-e 32.4%

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

   5. *-commutative 32.4%

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

7. Applied egg-rr 32.4%

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

8. Taylor expanded in x around -inf 0.0%

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

10. Simplified 36.6%

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

Alternative 7: 75.9% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 28.7%

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

5. Taylor expanded in t around inf 36.5%

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

   1. associate-*l* 36.5%

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

   2. +-commutative 36.5%

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

   3. sub-neg 36.5%

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

   4. metadata-eval 36.5%

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

   5. +-commutative 36.5%

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

7. Simplified 36.5%

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

8. Taylor expanded in x around inf 36.5%

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

9. Final simplification 36.5%

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

Alternative 8: 75.2% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Simplified 28.7%

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

5. Taylor expanded in t around inf 36.5%

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

   1. associate-*l* 36.5%

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

   2. +-commutative 36.5%

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

   3. sub-neg 36.5%

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

   4. metadata-eval 36.5%

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

   5. +-commutative 36.5%

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

7. Simplified 36.5%

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

8. Taylor expanded in x around inf 36.2%

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

Reproduce

herbie shell --seed 2024096 
(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)))))