Toniolo and Linder, Equation (7)

Percentage Accurate: 33.2% → 98.1%

Time: 20.2s

Alternatives: 10

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\mathsf{hypot}\left(\ell, t_1\right)}{\sqrt{x}}, t_1\right)}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= x -5.0)
     (/ t (fabs t))
     (/
      t
      (/ (hypot (* (sqrt 2.0) (/ (hypot l t_1) (sqrt x))) t_1) (sqrt 2.0))))))

C

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if (x <= -5.0) {
		tmp = t / fabs(t);
	} else {
		tmp = t / (hypot((sqrt(2.0) * (hypot(l, t_1) / sqrt(x))), t_1) / sqrt(2.0));
	}
	return tmp;
}

Java

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double tmp;
	if (x <= -5.0) {
		tmp = t / Math.abs(t);
	} else {
		tmp = t / (Math.hypot((Math.sqrt(2.0) * (Math.hypot(l, t_1) / Math.sqrt(x))), t_1) / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if x <= -5.0:
		tmp = t / math.fabs(t)
	else:
		tmp = t / (math.hypot((math.sqrt(2.0) * (math.hypot(l, t_1) / math.sqrt(x))), t_1) / math.sqrt(2.0))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(t / abs(t));
	else
		tmp = Float64(t / Float64(hypot(Float64(sqrt(2.0) * Float64(hypot(l, t_1) / sqrt(x))), t_1) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if (x <= -5.0)
		tmp = t / abs(t);
	else
		tmp = t / (hypot((sqrt(2.0) * (hypot(l, t_1) / sqrt(x))), t_1) / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.0], N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(t / N[(N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[l ^ 2 + t$95$1 ^ 2], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 43.2%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in x around inf 46.9%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 52.6%

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

      3. pow2 52.6%

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

      4. associate-/l* 52.7%

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

      5. sqrt-undiv 52.7%

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

      6. metadata-eval 52.7%

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

      7. metadata-eval 52.7%

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

   6. Applied egg-rr 52.7%

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

      1. /-rgt-identity 52.7%

         \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

      2. unpow2 52.7%

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

      3. rem-sqrt-square 98.7%

         \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   8. Simplified 98.7%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if -5 < x

   1. Initial program 29.9%

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

   3. Simplified 29.9%

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

   4. Taylor expanded in x around inf 53.9%

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

      1. add-sqr-sqrt 53.9%

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

      2. add-sqr-sqrt 53.9%

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

      3. hypot-def 53.9%

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

   6. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. *-commutative 97.6%

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

   8. Simplified 97.6%

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

4. Final simplification 98.1%

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

Alternative 2: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot {t}^{2}\\ t_2 := \frac{t}{\left|t\right|}\\ \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{t_1 + 2 \cdot \frac{t_1 + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow t 2.0))) (t_2 (/ t (fabs t))))
   (if (<= l 5.6e-26)
     t_2
     (if (<= l 3.8e+47)
       (/ t (/ (sqrt (+ t_1 (* 2.0 (/ (+ t_1 (pow l 2.0)) x)))) (sqrt 2.0)))
       (if (<= l 6.2e+168)
         t_2
         (/
          t
          (*
           (sqrt (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))
           (/ l (sqrt 2.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = 2.0 * pow(t, 2.0);
	double t_2 = t / fabs(t);
	double tmp;
	if (l <= 5.6e-26) {
		tmp = t_2;
	} else if (l <= 3.8e+47) {
		tmp = t / (sqrt((t_1 + (2.0 * ((t_1 + pow(l, 2.0)) / x)))) / sqrt(2.0));
	} else if (l <= 6.2e+168) {
		tmp = t_2;
	} else {
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))) * (l / sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * (t ** 2.0d0)
    t_2 = t / abs(t)
    if (l <= 5.6d-26) then
        tmp = t_2
    else if (l <= 3.8d+47) then
        tmp = t / (sqrt((t_1 + (2.0d0 * ((t_1 + (l ** 2.0d0)) / x)))) / sqrt(2.0d0))
    else if (l <= 6.2d+168) then
        tmp = t_2
    else
        tmp = t / (sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (1.0d0 / (x ** 2.0d0))))) * (l / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = 2.0 * Math.pow(t, 2.0);
	double t_2 = t / Math.abs(t);
	double tmp;
	if (l <= 5.6e-26) {
		tmp = t_2;
	} else if (l <= 3.8e+47) {
		tmp = t / (Math.sqrt((t_1 + (2.0 * ((t_1 + Math.pow(l, 2.0)) / x)))) / Math.sqrt(2.0));
	} else if (l <= 6.2e+168) {
		tmp = t_2;
	} else {
		tmp = t / (Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / Math.pow(x, 2.0))))) * (l / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = 2.0 * math.pow(t, 2.0)
	t_2 = t / math.fabs(t)
	tmp = 0
	if l <= 5.6e-26:
		tmp = t_2
	elif l <= 3.8e+47:
		tmp = t / (math.sqrt((t_1 + (2.0 * ((t_1 + math.pow(l, 2.0)) / x)))) / math.sqrt(2.0))
	elif l <= 6.2e+168:
		tmp = t_2
	else:
		tmp = t / (math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / math.pow(x, 2.0))))) * (l / math.sqrt(2.0)))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(2.0 * (t ^ 2.0))
	t_2 = Float64(t / abs(t))
	tmp = 0.0
	if (l <= 5.6e-26)
		tmp = t_2;
	elseif (l <= 3.8e+47)
		tmp = Float64(t / Float64(sqrt(Float64(t_1 + Float64(2.0 * Float64(Float64(t_1 + (l ^ 2.0)) / x)))) / sqrt(2.0)));
	elseif (l <= 6.2e+168)
		tmp = t_2;
	else
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0))))) * Float64(l / sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = 2.0 * (t ^ 2.0);
	t_2 = t / abs(t);
	tmp = 0.0;
	if (l <= 5.6e-26)
		tmp = t_2;
	elseif (l <= 3.8e+47)
		tmp = t / (sqrt((t_1 + (2.0 * ((t_1 + (l ^ 2.0)) / x)))) / sqrt(2.0));
	elseif (l <= 6.2e+168)
		tmp = t_2;
	else
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / (x ^ 2.0))))) * (l / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5.6e-26], t$95$2, If[LessEqual[l, 3.8e+47], N[(t / N[(N[Sqrt[N[(t$95$1 + N[(2.0 * N[(N[(t$95$1 + N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+168], t$95$2, N[(t / N[(N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.6000000000000002e-26 or 3.8000000000000003e47 < l < 6.19999999999999993e168

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in x around inf 41.8%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 45.8%

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

      3. pow2 45.8%

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

      4. associate-/l* 45.9%

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

      5. sqrt-undiv 45.9%

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

      6. metadata-eval 45.9%

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

      7. metadata-eval 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. /-rgt-identity 45.9%

         \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

      2. unpow2 45.9%

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

      3. rem-sqrt-square 81.1%

         \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   8. Simplified 81.1%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if 5.6000000000000002e-26 < l < 3.8000000000000003e47

   1. Initial program 40.6%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in x around inf 82.8%

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

   if 6.19999999999999993e168 < 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%

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

   4. Taylor expanded in l around inf 8.7%

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

      1. *-commutative 8.7%

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

      2. associate--l+ 26.6%

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

      3. sub-neg 26.6%

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

      4. metadata-eval 26.6%

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

      5. +-commutative 26.6%

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

      6. sub-neg 26.6%

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

      7. metadata-eval 26.6%

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

      8. +-commutative 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around inf 60.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \]

Alternative 3: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left|t\right|}\\ \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+168}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (fabs t))))
   (if (<= l 3.4e-25)
     t_1
     (if (<= l 5.5e+48)
       (/
        t
        (/
         (sqrt (+ (* 2.0 (pow t 2.0)) (* 2.0 (/ (pow l 2.0) x))))
         (sqrt 2.0)))
       (if (<= l 5.4e+168)
         t_1
         (/
          t
          (*
           (sqrt (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))
           (/ l (sqrt 2.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t / fabs(t);
	double tmp;
	if (l <= 3.4e-25) {
		tmp = t_1;
	} else if (l <= 5.5e+48) {
		tmp = t / (sqrt(((2.0 * pow(t, 2.0)) + (2.0 * (pow(l, 2.0) / x)))) / sqrt(2.0));
	} else if (l <= 5.4e+168) {
		tmp = t_1;
	} else {
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))) * (l / sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / abs(t)
    if (l <= 3.4d-25) then
        tmp = t_1
    else if (l <= 5.5d+48) then
        tmp = t / (sqrt(((2.0d0 * (t ** 2.0d0)) + (2.0d0 * ((l ** 2.0d0) / x)))) / sqrt(2.0d0))
    else if (l <= 5.4d+168) then
        tmp = t_1
    else
        tmp = t / (sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (1.0d0 / (x ** 2.0d0))))) * (l / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t / Math.abs(t);
	double tmp;
	if (l <= 3.4e-25) {
		tmp = t_1;
	} else if (l <= 5.5e+48) {
		tmp = t / (Math.sqrt(((2.0 * Math.pow(t, 2.0)) + (2.0 * (Math.pow(l, 2.0) / x)))) / Math.sqrt(2.0));
	} else if (l <= 5.4e+168) {
		tmp = t_1;
	} else {
		tmp = t / (Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / Math.pow(x, 2.0))))) * (l / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t / math.fabs(t)
	tmp = 0
	if l <= 3.4e-25:
		tmp = t_1
	elif l <= 5.5e+48:
		tmp = t / (math.sqrt(((2.0 * math.pow(t, 2.0)) + (2.0 * (math.pow(l, 2.0) / x)))) / math.sqrt(2.0))
	elif l <= 5.4e+168:
		tmp = t_1
	else:
		tmp = t / (math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / math.pow(x, 2.0))))) * (l / math.sqrt(2.0)))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t / abs(t))
	tmp = 0.0
	if (l <= 3.4e-25)
		tmp = t_1;
	elseif (l <= 5.5e+48)
		tmp = Float64(t / Float64(sqrt(Float64(Float64(2.0 * (t ^ 2.0)) + Float64(2.0 * Float64((l ^ 2.0) / x)))) / sqrt(2.0)));
	elseif (l <= 5.4e+168)
		tmp = t_1;
	else
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0))))) * Float64(l / sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t / abs(t);
	tmp = 0.0;
	if (l <= 3.4e-25)
		tmp = t_1;
	elseif (l <= 5.5e+48)
		tmp = t / (sqrt(((2.0 * (t ^ 2.0)) + (2.0 * ((l ^ 2.0) / x)))) / sqrt(2.0));
	elseif (l <= 5.4e+168)
		tmp = t_1;
	else
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / (x ^ 2.0))))) * (l / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 3.4e-25], t$95$1, If[LessEqual[l, 5.5e+48], N[(t / N[(N[Sqrt[N[(N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e+168], t$95$1, N[(t / N[(N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.40000000000000002e-25 or 5.5000000000000002e48 < l < 5.40000000000000031e168

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in x around inf 41.8%

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

      1. add-sqr-sqrt 40.6%

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

      2. sqrt-unprod 45.8%

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

      3. pow2 45.8%

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

      4. associate-/l* 45.9%

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

      5. sqrt-undiv 45.9%

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

      6. metadata-eval 45.9%

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

      7. metadata-eval 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. /-rgt-identity 45.9%

         \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

      2. unpow2 45.9%

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

      3. rem-sqrt-square 81.1%

         \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   8. Simplified 81.1%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if 3.40000000000000002e-25 < l < 5.5000000000000002e48

   1. Initial program 40.6%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in x around inf 82.8%

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

   5. Taylor expanded in t around 0 83.1%

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

   if 5.40000000000000031e168 < 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%

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

   4. Taylor expanded in l around inf 8.7%

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

      1. *-commutative 8.7%

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

      2. associate--l+ 26.6%

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

      3. sub-neg 26.6%

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

      4. metadata-eval 26.6%

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

      5. +-commutative 26.6%

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

      6. sub-neg 26.6%

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

      7. metadata-eval 26.6%

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

      8. +-commutative 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around inf 60.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{x}}}{\sqrt{2}}}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \]

Alternative 4: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= l 5.4e+168)
   (/ t (fabs t))
   (/
    t
    (*
     (sqrt (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 x) (/ 1.0 (pow x 2.0)))))
     (/ l (sqrt 2.0))))))

C

double code(double x, double l, double t) {
	double tmp;
	if (l <= 5.4e+168) {
		tmp = t / fabs(t);
	} else {
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / pow(x, 2.0))))) * (l / sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (l <= 5.4d+168) then
        tmp = t / abs(t)
    else
        tmp = t / (sqrt(((1.0d0 / (x + (-1.0d0))) + ((1.0d0 / x) + (1.0d0 / (x ** 2.0d0))))) * (l / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (l <= 5.4e+168) {
		tmp = t / Math.abs(t);
	} else {
		tmp = t / (Math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / Math.pow(x, 2.0))))) * (l / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if l <= 5.4e+168:
		tmp = t / math.fabs(t)
	else:
		tmp = t / (math.sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / math.pow(x, 2.0))))) * (l / math.sqrt(2.0)))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (l <= 5.4e+168)
		tmp = Float64(t / abs(t));
	else
		tmp = Float64(t / Float64(sqrt(Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / x) + Float64(1.0 / (x ^ 2.0))))) * Float64(l / sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (l <= 5.4e+168)
		tmp = t / abs(t);
	else
		tmp = t / (sqrt(((1.0 / (x + -1.0)) + ((1.0 / x) + (1.0 / (x ^ 2.0))))) * (l / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[l, 5.4e+168], N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(t / N[(N[Sqrt[N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.40000000000000031e168

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in x around inf 40.3%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 45.8%

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

      3. pow2 45.8%

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

      4. associate-/l* 45.9%

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

      5. sqrt-undiv 45.9%

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

      6. metadata-eval 45.9%

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

      7. metadata-eval 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. /-rgt-identity 45.9%

         \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

      2. unpow2 45.9%

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

      3. rem-sqrt-square 79.7%

         \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   8. Simplified 79.7%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if 5.40000000000000031e168 < 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%

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

   4. Taylor expanded in l around inf 8.7%

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

      1. *-commutative 8.7%

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

      2. associate--l+ 26.6%

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

      3. sub-neg 26.6%

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

      4. metadata-eval 26.6%

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

      5. +-commutative 26.6%

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

      6. sub-neg 26.6%

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

      7. metadata-eval 26.6%

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

      8. +-commutative 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around inf 60.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x + -1} + \left(\frac{1}{x} + \frac{1}{{x}^{2}}\right)} \cdot \frac{\ell}{\sqrt{2}}}\\ \end{array} \]

Alternative 5: 77.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= l 5.5e+168) (/ t (fabs t)) (* t (/ (sqrt x) l))))

C

double code(double x, double l, double t) {
	double tmp;
	if (l <= 5.5e+168) {
		tmp = t / fabs(t);
	} else {
		tmp = t * (sqrt(x) / l);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (l <= 5.5d+168) then
        tmp = t / abs(t)
    else
        tmp = t * (sqrt(x) / l)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (l <= 5.5e+168) {
		tmp = t / Math.abs(t);
	} else {
		tmp = t * (Math.sqrt(x) / l);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if l <= 5.5e+168:
		tmp = t / math.fabs(t)
	else:
		tmp = t * (math.sqrt(x) / l)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (l <= 5.5e+168)
		tmp = Float64(t / abs(t));
	else
		tmp = Float64(t * Float64(sqrt(x) / l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (l <= 5.5e+168)
		tmp = t / abs(t);
	else
		tmp = t * (sqrt(x) / l);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[l, 5.5e+168], N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.5000000000000001e168

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   4. Taylor expanded in x around inf 40.3%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 45.8%

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

      3. pow2 45.8%

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

      4. associate-/l* 45.9%

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

      5. sqrt-undiv 45.9%

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

      6. metadata-eval 45.9%

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

      7. metadata-eval 45.9%

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

   6. Applied egg-rr 45.9%

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

      1. /-rgt-identity 45.9%

         \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

      2. unpow2 45.9%

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

      3. rem-sqrt-square 79.7%

         \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   8. Simplified 79.7%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

   if 5.5000000000000001e168 < 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%

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

   4. Taylor expanded in l around inf 8.3%

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

      1. *-commutative 8.3%

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

      2. associate--l+ 26.0%

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

      3. sub-neg 26.0%

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

      4. metadata-eval 26.0%

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

      5. +-commutative 26.0%

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

      6. sub-neg 26.0%

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

      7. metadata-eval 26.0%

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

      8. +-commutative 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in x around inf 51.8%

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

      1. *-commutative 51.8%

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

   9. Simplified 51.8%

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

       1. expm1-log1p-u 50.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\right)\right)} \]

       2. expm1-udef 16.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{x \cdot 0.5} \cdot \frac{t}{\ell}\right)\right)} - 1} \]

       3. associate-*r* 16.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{x \cdot 0.5}\right) \cdot \frac{t}{\ell}}\right)} - 1 \]

       4. *-commutative 16.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot \left(\sqrt{2} \cdot \sqrt{x \cdot 0.5}\right)}\right)} - 1 \]

       5. sqrt-unprod 16.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{t}{\ell} \cdot \color{blue}{\sqrt{2 \cdot \left(x \cdot 0.5\right)}}\right)} - 1 \]

   11. Applied egg-rr 16.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{t}{\ell} \cdot \sqrt{2 \cdot \left(x \cdot 0.5\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 51.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot \sqrt{2 \cdot \left(x \cdot 0.5\right)}\right)\right)} \]

       2. expm1-log1p 52.0%

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

       3. associate-*l/ 59.2%

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

       4. *-commutative 59.2%

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

       5. associate-*l/ 59.5%

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

       6. *-commutative 59.5%

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

       7. *-commutative 59.5%

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

       8. associate-*l* 59.5%

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

       9. metadata-eval 59.5%

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

   13. Simplified 59.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{t}{\left|t\right|}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 6: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\left|t\right|} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (/ t (fabs t)))

C

double code(double x, double l, double t) {
	return t / fabs(t);
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = t / abs(t)
end function

Java

public static double code(double x, double l, double t) {
	return t / Math.abs(t);
}

Python

def code(x, l, t):
	return t / math.fabs(t)

Julia

function code(x, l, t)
	return Float64(t / abs(t))
end

MATLAB

function tmp = code(x, l, t)
	tmp = t / abs(t);
end

Wolfram

code[x_, l_, t_] := N[(t / N[Abs[t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Simplified 35.5%

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

4. Taylor expanded in x around inf 37.8%

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

   1. add-sqr-sqrt 36.6%

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

   2. sqrt-unprod 42.5%

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

   3. pow2 42.5%

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

   4. associate-/l* 42.6%

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

   5. sqrt-undiv 42.6%

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

   6. metadata-eval 42.6%

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

   7. metadata-eval 42.6%

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

6. Applied egg-rr 42.6%

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

   1. /-rgt-identity 42.6%

      \[\leadsto \frac{t}{\sqrt{{\color{blue}{t}}^{2}}} \]

   2. unpow2 42.6%

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

   3. rem-sqrt-square 76.3%

      \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

8. Simplified 76.3%

   \[\leadsto \frac{t}{\color{blue}{\left|t\right|}} \]

9. Final simplification 76.3%

   \[\leadsto \frac{t}{\left|t\right|} \]

Alternative 7: 75.5% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -2e-310) (+ -1.0 (/ 1.0 x)) 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 75.9%

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

   9. Simplified 75.9%

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

   if -1.999999999999994e-310 < t

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in t around inf 78.4%

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

   7. Taylor expanded in x around inf 77.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 75.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310) (+ -1.0 (/ 1.0 x)) (+ 1.0 (/ -1.0 x))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2d-310)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 75.9%

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

   9. Simplified 75.9%

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

   if -1.999999999999994e-310 < t

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in t around inf 78.4%

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

   7. Taylor expanded in x around inf 78.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 9: 75.1% accurate, 73.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 (if (<= t -2e-310) -1.0 1.0))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 30.6%

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

   3. Simplified 30.5%

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

   5. Applied egg-rr 64.2%

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

   6. Taylor expanded in t around inf 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

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

      2. rem-square-sqrt 75.6%

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

   9. Simplified 75.6%

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

   if -1.999999999999994e-310 < t

   1. Initial program 40.8%

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

   3. Simplified 40.8%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in t around inf 78.4%

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

   7. Taylor expanded in x around inf 77.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 38.3% accurate, 225.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x l t) :precision binary64 -1.0)

C

double code(double x, double l, double t) {
	return -1.0;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

public static double code(double x, double l, double t) {
	return -1.0;
}

Python

def code(x, l, t):
	return -1.0

Julia

function code(x, l, t)
	return -1.0
end

MATLAB

function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

code[x_, l_, t_] := -1.0

Derivation

1. Initial program 35.5%

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

3. Simplified 35.4%

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

5. Applied egg-rr 68.8%

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

6. Taylor expanded in t around inf 38.6%

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

7. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 0.0%

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

   2. rem-square-sqrt 40.1%

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

9. Simplified 40.1%

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

10. Final simplification 40.1%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023307 
(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)))))