Toniolo and Linder, Equation (7)

Percentage Accurate: 34.4% → 84.9%

Time: 15.9s

Alternatives: 9

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: 34.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := \frac{t}{\sqrt{0.5 \cdot \left(t_1 + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, t_1\right)\right)\right)}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.46 \cdot 10^{+145}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-302}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* l (/ l x)))
        (t_2
         (/
          t
          (sqrt
           (* 0.5 (+ t_1 (fma 2.0 (/ t (/ x t)) (fma 2.0 (* t t) t_1)))))))
        (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.46e+145)
     (/ t (- t))
     (if (<= t -2.7e-197)
       t_2
       (if (<= t -1e-302) (- t_3) (if (<= t 1.15e+23) t_2 t_3))))))

C

double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t / sqrt((0.5 * (t_1 + fma(2.0, (t / (x / t)), fma(2.0, (t * t), t_1)))));
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.46e+145) {
		tmp = t / -t;
	} else if (t <= -2.7e-197) {
		tmp = t_2;
	} else if (t <= -1e-302) {
		tmp = -t_3;
	} else if (t <= 1.15e+23) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t / sqrt(Float64(0.5 * Float64(t_1 + fma(2.0, Float64(t / Float64(x / t)), fma(2.0, Float64(t * t), t_1))))))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.46e+145)
		tmp = Float64(t / Float64(-t));
	elseif (t <= -2.7e-197)
		tmp = t_2;
	elseif (t <= -1e-302)
		tmp = Float64(-t_3);
	elseif (t <= 1.15e+23)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(0.5 * N[(t$95$1 + N[(2.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.46e+145], N[(t / (-t)), $MachinePrecision], If[LessEqual[t, -2.7e-197], t$95$2, If[LessEqual[t, -1e-302], (-t$95$3), If[LessEqual[t, 1.15e+23], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.45999999999999988e145

   1. Initial program 7.1%

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

      1. associate-*l/ 7.1%

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

   3. Simplified 7.1%

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

      1. *-commutative 7.1%

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

      2. clear-num 7.2%

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

      3. un-div-inv 7.1%

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

      4. sqrt-undiv 7.2%

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

      5. metadata-eval 7.2%

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

      6. sub-neg 7.2%

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

      7. associate-*l/ 2.4%

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

      8. sub-neg 2.4%

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

      9. metadata-eval 2.4%

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

   5. Applied egg-rr 2.4%

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

      1. +-commutative 2.4%

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

      2. *-commutative 2.4%

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

      3. +-commutative 2.4%

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

      4. +-commutative 2.4%

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

   7. Simplified 2.4%

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

   8. Taylor expanded in x around inf 7.9%

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

      1. unpow2 7.9%

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

   10. Simplified 7.9%

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

   11. Taylor expanded in t around -inf 97.7%

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

       1. mul-1-neg 97.7%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 97.7%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.45999999999999988e145 < t < -2.70000000000000017e-197 or -9.9999999999999996e-303 < t < 1.15e23

   1. Initial program 42.5%

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

      1. associate-*l/ 42.5%

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

   3. Simplified 42.5%

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

      1. *-commutative 42.5%

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

      2. clear-num 42.5%

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

      3. un-div-inv 42.5%

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

      4. sqrt-undiv 42.7%

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

      5. metadata-eval 42.7%

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

      6. sub-neg 42.7%

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

      7. associate-*l/ 34.1%

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

      8. sub-neg 34.1%

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

      9. metadata-eval 34.1%

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

   5. Applied egg-rr 34.1%

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

      1. +-commutative 34.1%

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

      2. *-commutative 34.1%

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

      3. +-commutative 34.1%

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

      4. +-commutative 34.1%

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

   7. Simplified 34.1%

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

   8. Taylor expanded in x around inf 78.4%

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

      1. cancel-sign-sub-inv 78.4%

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

      2. fma-def 78.4%

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

      3. unpow2 78.4%

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

      4. fma-def 78.4%

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

      5. unpow2 78.4%

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

      6. unpow2 78.4%

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

      7. metadata-eval 78.4%

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

      8. unpow2 78.4%

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

      9. unpow2 78.4%

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

      10. fma-udef 78.4%

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

      11. *-lft-identity 78.4%

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

      12. fma-udef 78.4%

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

      13. unpow2 78.4%

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

      14. unpow2 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in l around inf 78.0%

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

       1. unpow2 78.0%

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

   13. Simplified 78.0%

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

       1. expm1-log1p-u 75.6%

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

       2. expm1-udef 41.9%

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

   15. Applied egg-rr 50.6%

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

       1. expm1-def 84.3%

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

       2. expm1-log1p 87.3%

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

       3. *-commutative 87.3%

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

       4. associate-/r/ 87.4%

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

       5. associate-/l* 87.4%

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

       6. associate-/r/ 87.3%

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

   17. Simplified 87.3%

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

   if -2.70000000000000017e-197 < t < -9.9999999999999996e-303

   1. Initial program 6.8%

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

      1. associate-*l/ 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in t around inf 2.8%

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

      1. +-commutative 2.8%

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

      2. associate-*r/ 2.8%

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

      3. sub-neg 2.8%

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

      4. metadata-eval 2.8%

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

      5. unpow2 2.8%

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

      6. +-commutative 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in t around -inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. sub-neg 64.3%

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

      3. metadata-eval 64.3%

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

   9. Simplified 64.3%

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

   if 1.15e23 < t

   1. Initial program 33.3%

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

      1. associate-*l/ 33.4%

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

   3. Simplified 33.4%

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

   4. Taylor expanded in t around inf 17.9%

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

      1. +-commutative 17.9%

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

      2. associate-*r/ 40.9%

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

      3. sub-neg 40.9%

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

      4. metadata-eval 40.9%

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

      5. unpow2 40.9%

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

      6. +-commutative 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in t around 0 96.9%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{+145}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-302}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+23}:\\ \;\;\;\;\frac{t}{\sqrt{0.5 \cdot \left(\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-t}\\ t_2 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (- t))) (t_2 (/ t (/ l (sqrt x)))))
   (if (<= t -5.2e-104)
     t_1
     (if (<= t -4.5e-197)
       t_2
       (if (<= t -1.8e-302)
         t_1
         (if (<= t 1.35e-140) t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = t / (l / sqrt(x));
	double tmp;
	if (t <= -5.2e-104) {
		tmp = t_1;
	} else if (t <= -4.5e-197) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 1.35e-140) {
		tmp = t_2;
	} else {
		tmp = sqrt(((x + -1.0) / (x + 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t / -t
    t_2 = t / (l / sqrt(x))
    if (t <= (-5.2d-104)) then
        tmp = t_1
    else if (t <= (-4.5d-197)) then
        tmp = t_2
    else if (t <= (-1.8d-302)) then
        tmp = t_1
    else if (t <= 1.35d-140) then
        tmp = t_2
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = t / (l / Math.sqrt(x));
	double tmp;
	if (t <= -5.2e-104) {
		tmp = t_1;
	} else if (t <= -4.5e-197) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 1.35e-140) {
		tmp = t_2;
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t / -t
	t_2 = t / (l / math.sqrt(x))
	tmp = 0
	if t <= -5.2e-104:
		tmp = t_1
	elif t <= -4.5e-197:
		tmp = t_2
	elif t <= -1.8e-302:
		tmp = t_1
	elif t <= 1.35e-140:
		tmp = t_2
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t / Float64(-t))
	t_2 = Float64(t / Float64(l / sqrt(x)))
	tmp = 0.0
	if (t <= -5.2e-104)
		tmp = t_1;
	elseif (t <= -4.5e-197)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 1.35e-140)
		tmp = t_2;
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t / -t;
	t_2 = t / (l / sqrt(x));
	tmp = 0.0;
	if (t <= -5.2e-104)
		tmp = t_1;
	elseif (t <= -4.5e-197)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 1.35e-140)
		tmp = t_2;
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e-104], t$95$1, If[LessEqual[t, -4.5e-197], t$95$2, If[LessEqual[t, -1.8e-302], t$95$1, If[LessEqual[t, 1.35e-140], t$95$2, N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.20000000000000005e-104 or -4.5000000000000001e-197 < t < -1.8e-302

   1. Initial program 32.7%

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

      1. associate-*l/ 32.7%

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

   3. Simplified 32.7%

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

      1. *-commutative 32.7%

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

      2. clear-num 32.7%

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

      3. un-div-inv 32.7%

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

      4. sqrt-undiv 32.9%

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

      5. metadata-eval 32.9%

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

      6. sub-neg 32.9%

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

      7. associate-*l/ 20.3%

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

      8. sub-neg 20.3%

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

      9. metadata-eval 20.3%

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

   5. Applied egg-rr 20.3%

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

      1. +-commutative 20.3%

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

      2. *-commutative 20.3%

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

      3. +-commutative 20.3%

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

      4. +-commutative 20.3%

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

   7. Simplified 20.3%

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

   8. Taylor expanded in x around inf 37.1%

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

      1. unpow2 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in t around -inf 82.0%

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

       1. mul-1-neg 82.0%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 82.0%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -5.20000000000000005e-104 < t < -4.5000000000000001e-197 or -1.8e-302 < t < 1.35e-140

   1. Initial program 10.6%

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

      1. associate-*l/ 10.6%

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

   3. Simplified 10.6%

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

      1. *-commutative 10.6%

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

      2. clear-num 10.6%

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

      3. un-div-inv 10.6%

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

      4. sqrt-undiv 10.6%

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

      5. metadata-eval 10.6%

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

      6. sub-neg 10.6%

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

      7. associate-*l/ 13.2%

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

      8. sub-neg 13.2%

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

      9. metadata-eval 13.2%

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

   5. Applied egg-rr 13.2%

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

      1. +-commutative 13.2%

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

      2. *-commutative 13.2%

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

      3. +-commutative 13.2%

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

      4. +-commutative 13.2%

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

   7. Simplified 13.2%

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

   8. Taylor expanded in x around inf 67.5%

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

      1. cancel-sign-sub-inv 67.5%

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

      2. fma-def 67.5%

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

      3. unpow2 67.5%

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

      4. fma-def 67.5%

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

      5. unpow2 67.5%

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

      6. unpow2 67.5%

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

      7. metadata-eval 67.5%

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

      8. unpow2 67.5%

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

      9. unpow2 67.5%

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

      10. fma-udef 67.5%

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

      11. *-lft-identity 67.5%

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

      12. fma-udef 67.5%

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

      13. unpow2 67.5%

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

      14. unpow2 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in t around 0 47.9%

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

       1. associate-*l* 47.9%

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

       2. *-commutative 47.9%

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

       3. associate-*l* 48.0%

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

   13. Simplified 48.0%

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

       1. expm1-log1p-u 39.1%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)\right)}} \]

       2. expm1-udef 24.9%

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

       3. associate-*r* 24.9%

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

       4. sqrt-unprod 24.9%

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

       5. metadata-eval 24.9%

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

       6. metadata-eval 24.9%

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

       7. *-un-lft-identity 24.9%

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

       8. add-sqr-sqrt 24.3%

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

       9. sqrt-prod 35.6%

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

       10. sqrt-prod 35.9%

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

       11. div-inv 35.9%

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

       12. sqrt-div 35.6%

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

       13. sqrt-prod 24.3%

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

       14. add-sqr-sqrt 24.9%

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

   15. Applied egg-rr 24.9%

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

       1. expm1-def 39.2%

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

       2. expm1-log1p 48.2%

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

   17. Simplified 48.2%

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

   if 1.35e-140 < t

   1. Initial program 38.4%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in t around inf 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-*r/ 49.9%

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

      3. sub-neg 49.9%

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

      4. metadata-eval 49.9%

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

      5. unpow2 49.9%

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

      6. +-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in t around 0 83.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (/ l (sqrt x))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_3 (- t_2)))
   (if (<= t -5.2e-104)
     t_3
     (if (<= t -7e-197)
       t_1
       (if (<= t -1.8e-302) t_3 (if (<= t 1.8e-142) t_1 t_2))))))

C

double code(double x, double l, double t) {
	double t_1 = t / (l / sqrt(x));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = -t_2;
	double tmp;
	if (t <= -5.2e-104) {
		tmp = t_3;
	} else if (t <= -7e-197) {
		tmp = t_1;
	} else if (t <= -1.8e-302) {
		tmp = t_3;
	} else if (t <= 1.8e-142) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = t / (l / sqrt(x))
    t_2 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_3 = -t_2
    if (t <= (-5.2d-104)) then
        tmp = t_3
    else if (t <= (-7d-197)) then
        tmp = t_1
    else if (t <= (-1.8d-302)) then
        tmp = t_3
    else if (t <= 1.8d-142) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t / (l / Math.sqrt(x));
	double t_2 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = -t_2;
	double tmp;
	if (t <= -5.2e-104) {
		tmp = t_3;
	} else if (t <= -7e-197) {
		tmp = t_1;
	} else if (t <= -1.8e-302) {
		tmp = t_3;
	} else if (t <= 1.8e-142) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t / (l / math.sqrt(x))
	t_2 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_3 = -t_2
	tmp = 0
	if t <= -5.2e-104:
		tmp = t_3
	elif t <= -7e-197:
		tmp = t_1
	elif t <= -1.8e-302:
		tmp = t_3
	elif t <= 1.8e-142:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t / Float64(l / sqrt(x)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_3 = Float64(-t_2)
	tmp = 0.0
	if (t <= -5.2e-104)
		tmp = t_3;
	elseif (t <= -7e-197)
		tmp = t_1;
	elseif (t <= -1.8e-302)
		tmp = t_3;
	elseif (t <= 1.8e-142)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t / (l / sqrt(x));
	t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	t_3 = -t_2;
	tmp = 0.0;
	if (t <= -5.2e-104)
		tmp = t_3;
	elseif (t <= -7e-197)
		tmp = t_1;
	elseif (t <= -1.8e-302)
		tmp = t_3;
	elseif (t <= 1.8e-142)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, If[LessEqual[t, -5.2e-104], t$95$3, If[LessEqual[t, -7e-197], t$95$1, If[LessEqual[t, -1.8e-302], t$95$3, If[LessEqual[t, 1.8e-142], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.20000000000000005e-104 or -6.9999999999999996e-197 < t < -1.8e-302

   1. Initial program 32.7%

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

      1. associate-*l/ 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in t around inf 22.6%

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

      1. +-commutative 22.6%

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

      2. associate-*r/ 37.8%

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

      3. sub-neg 37.8%

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

      4. metadata-eval 37.8%

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

      5. unpow2 37.8%

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

      6. +-commutative 37.8%

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

   6. Simplified 37.8%

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

   7. Taylor expanded in t around -inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. sub-neg 82.9%

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

      3. metadata-eval 82.9%

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

   9. Simplified 82.9%

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

   if -5.20000000000000005e-104 < t < -6.9999999999999996e-197 or -1.8e-302 < t < 1.8e-142

   1. Initial program 10.6%

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

      1. associate-*l/ 10.6%

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

   3. Simplified 10.6%

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

      1. *-commutative 10.6%

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

      2. clear-num 10.6%

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

      3. un-div-inv 10.6%

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

      4. sqrt-undiv 10.6%

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

      5. metadata-eval 10.6%

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

      6. sub-neg 10.6%

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

      7. associate-*l/ 13.2%

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

      8. sub-neg 13.2%

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

      9. metadata-eval 13.2%

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

   5. Applied egg-rr 13.2%

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

      1. +-commutative 13.2%

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

      2. *-commutative 13.2%

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

      3. +-commutative 13.2%

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

      4. +-commutative 13.2%

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

   7. Simplified 13.2%

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

   8. Taylor expanded in x around inf 67.5%

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

      1. cancel-sign-sub-inv 67.5%

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

      2. fma-def 67.5%

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

      3. unpow2 67.5%

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

      4. fma-def 67.5%

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

      5. unpow2 67.5%

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

      6. unpow2 67.5%

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

      7. metadata-eval 67.5%

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

      8. unpow2 67.5%

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

      9. unpow2 67.5%

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

      10. fma-udef 67.5%

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

      11. *-lft-identity 67.5%

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

      12. fma-udef 67.5%

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

      13. unpow2 67.5%

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

      14. unpow2 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in t around 0 47.9%

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

       1. associate-*l* 47.9%

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

       2. *-commutative 47.9%

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

       3. associate-*l* 48.0%

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

   13. Simplified 48.0%

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

       1. expm1-log1p-u 39.1%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)\right)}} \]

       2. expm1-udef 24.9%

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

       3. associate-*r* 24.9%

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

       4. sqrt-unprod 24.9%

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

       5. metadata-eval 24.9%

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

       6. metadata-eval 24.9%

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

       7. *-un-lft-identity 24.9%

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

       8. add-sqr-sqrt 24.3%

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

       9. sqrt-prod 35.6%

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

       10. sqrt-prod 35.9%

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

       11. div-inv 35.9%

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

       12. sqrt-div 35.6%

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

       13. sqrt-prod 24.3%

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

       14. add-sqr-sqrt 24.9%

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

   15. Applied egg-rr 24.9%

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

       1. expm1-def 39.2%

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

       2. expm1-log1p 48.2%

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

   17. Simplified 48.2%

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

   if 1.8e-142 < t

   1. Initial program 38.4%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in t around inf 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-*r/ 49.9%

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

      3. sub-neg 49.9%

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

      4. metadata-eval 49.9%

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

      5. unpow2 49.9%

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

      6. +-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in t around 0 83.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-197}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-t}\\ t_2 := \sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (- t))) (t_2 (* (sqrt x) (/ t l))))
   (if (<= t -1.02e-145)
     t_1
     (if (<= t -4.1e-197)
       t_2
       (if (<= t -1.8e-302)
         t_1
         (if (<= t 1.36e-142) t_2 (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = sqrt(x) * (t / l);
	double tmp;
	if (t <= -1.02e-145) {
		tmp = t_1;
	} else if (t <= -4.1e-197) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 1.36e-142) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t / -t
    t_2 = sqrt(x) * (t / l)
    if (t <= (-1.02d-145)) then
        tmp = t_1
    else if (t <= (-4.1d-197)) then
        tmp = t_2
    else if (t <= (-1.8d-302)) then
        tmp = t_1
    else if (t <= 1.36d-142) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = Math.sqrt(x) * (t / l);
	double tmp;
	if (t <= -1.02e-145) {
		tmp = t_1;
	} else if (t <= -4.1e-197) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 1.36e-142) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t / -t
	t_2 = math.sqrt(x) * (t / l)
	tmp = 0
	if t <= -1.02e-145:
		tmp = t_1
	elif t <= -4.1e-197:
		tmp = t_2
	elif t <= -1.8e-302:
		tmp = t_1
	elif t <= 1.36e-142:
		tmp = t_2
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t / Float64(-t))
	t_2 = Float64(sqrt(x) * Float64(t / l))
	tmp = 0.0
	if (t <= -1.02e-145)
		tmp = t_1;
	elseif (t <= -4.1e-197)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 1.36e-142)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t / -t;
	t_2 = sqrt(x) * (t / l);
	tmp = 0.0;
	if (t <= -1.02e-145)
		tmp = t_1;
	elseif (t <= -4.1e-197)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 1.36e-142)
		tmp = t_2;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.02e-145], t$95$1, If[LessEqual[t, -4.1e-197], t$95$2, If[LessEqual[t, -1.8e-302], t$95$1, If[LessEqual[t, 1.36e-142], t$95$2, N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.01999999999999993e-145 or -4.1e-197 < t < -1.8e-302

   1. Initial program 32.3%

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

      1. associate-*l/ 32.3%

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

   3. Simplified 32.3%

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

      1. *-commutative 32.3%

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

      2. clear-num 32.3%

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

      3. un-div-inv 32.3%

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

      4. sqrt-undiv 32.4%

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

      5. metadata-eval 32.4%

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

      6. sub-neg 32.4%

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

      7. associate-*l/ 20.6%

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

      8. sub-neg 20.6%

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

      9. metadata-eval 20.6%

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

   5. Applied egg-rr 20.6%

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

      1. +-commutative 20.6%

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

      2. *-commutative 20.6%

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

      3. +-commutative 20.6%

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

      4. +-commutative 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in x around inf 37.6%

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

      1. unpow2 37.6%

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

   10. Simplified 37.6%

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

   11. Taylor expanded in t around -inf 79.9%

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

       1. mul-1-neg 79.9%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 79.9%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.01999999999999993e-145 < t < -4.1e-197 or -1.8e-302 < t < 1.35999999999999993e-142

   1. Initial program 7.5%

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

      1. associate-*l/ 7.5%

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

   3. Simplified 7.5%

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

      1. *-commutative 7.5%

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

      2. clear-num 7.5%

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

      3. un-div-inv 7.5%

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

      4. sqrt-undiv 7.5%

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

      5. metadata-eval 7.5%

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

      6. sub-neg 7.5%

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

      7. associate-*l/ 10.5%

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

      8. sub-neg 10.5%

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

      9. metadata-eval 10.5%

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

   5. Applied egg-rr 10.5%

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

      1. +-commutative 10.5%

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

      2. *-commutative 10.5%

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

      3. +-commutative 10.5%

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

      4. +-commutative 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in x around inf 68.7%

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

      1. cancel-sign-sub-inv 68.7%

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

      2. fma-def 68.7%

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

      3. unpow2 68.7%

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

      4. fma-def 68.7%

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

      5. unpow2 68.7%

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

      6. unpow2 68.7%

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

      7. metadata-eval 68.7%

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

      8. unpow2 68.7%

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

      9. unpow2 68.7%

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

      10. fma-udef 68.7%

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

      11. *-lft-identity 68.7%

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

      12. fma-udef 68.7%

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

      13. unpow2 68.7%

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

      14. unpow2 68.7%

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

   10. Simplified 68.7%

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

   11. Taylor expanded in l around inf 68.7%

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

       1. unpow2 68.7%

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

   13. Simplified 68.7%

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

   14. Taylor expanded in t around 0 49.0%

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

   if 1.35999999999999993e-142 < t

   1. Initial program 38.4%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in t around inf 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-*r/ 49.9%

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

      3. sub-neg 49.9%

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

      4. metadata-eval 49.9%

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

      5. unpow2 49.9%

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

      6. +-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in x around inf 82.3%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   8. Step-by-step derivation

      1. associate--l+ 82.3%

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

      2. associate-*r/ 82.3%

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

      3. metadata-eval 82.3%

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

      4. unpow2 82.3%

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

   9. Simplified 82.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-145}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5: 74.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-t}\\ t_2 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (- t))) (t_2 (/ t (/ l (sqrt x)))))
   (if (<= t -6.6e-104)
     t_1
     (if (<= t -1.02e-196)
       t_2
       (if (<= t -1.8e-302)
         t_1
         (if (<= t 6e-140) t_2 (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))))))

C

double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = t / (l / sqrt(x));
	double tmp;
	if (t <= -6.6e-104) {
		tmp = t_1;
	} else if (t <= -1.02e-196) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 6e-140) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t / -t
    t_2 = t / (l / sqrt(x))
    if (t <= (-6.6d-104)) then
        tmp = t_1
    else if (t <= (-1.02d-196)) then
        tmp = t_2
    else if (t <= (-1.8d-302)) then
        tmp = t_1
    else if (t <= 6d-140) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double l, double t) {
	double t_1 = t / -t;
	double t_2 = t / (l / Math.sqrt(x));
	double tmp;
	if (t <= -6.6e-104) {
		tmp = t_1;
	} else if (t <= -1.02e-196) {
		tmp = t_2;
	} else if (t <= -1.8e-302) {
		tmp = t_1;
	} else if (t <= 6e-140) {
		tmp = t_2;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	t_1 = t / -t
	t_2 = t / (l / math.sqrt(x))
	tmp = 0
	if t <= -6.6e-104:
		tmp = t_1
	elif t <= -1.02e-196:
		tmp = t_2
	elif t <= -1.8e-302:
		tmp = t_1
	elif t <= 6e-140:
		tmp = t_2
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	t_1 = Float64(t / Float64(-t))
	t_2 = Float64(t / Float64(l / sqrt(x)))
	tmp = 0.0
	if (t <= -6.6e-104)
		tmp = t_1;
	elseif (t <= -1.02e-196)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 6e-140)
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	t_1 = t / -t;
	t_2 = t / (l / sqrt(x));
	tmp = 0.0;
	if (t <= -6.6e-104)
		tmp = t_1;
	elseif (t <= -1.02e-196)
		tmp = t_2;
	elseif (t <= -1.8e-302)
		tmp = t_1;
	elseif (t <= 6e-140)
		tmp = t_2;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := Block[{t$95$1 = N[(t / (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.6e-104], t$95$1, If[LessEqual[t, -1.02e-196], t$95$2, If[LessEqual[t, -1.8e-302], t$95$1, If[LessEqual[t, 6e-140], t$95$2, N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.60000000000000004e-104 or -1.0200000000000001e-196 < t < -1.8e-302

   1. Initial program 32.7%

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

      1. associate-*l/ 32.7%

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

   3. Simplified 32.7%

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

      1. *-commutative 32.7%

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

      2. clear-num 32.7%

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

      3. un-div-inv 32.7%

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

      4. sqrt-undiv 32.9%

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

      5. metadata-eval 32.9%

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

      6. sub-neg 32.9%

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

      7. associate-*l/ 20.3%

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

      8. sub-neg 20.3%

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

      9. metadata-eval 20.3%

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

   5. Applied egg-rr 20.3%

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

      1. +-commutative 20.3%

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

      2. *-commutative 20.3%

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

      3. +-commutative 20.3%

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

      4. +-commutative 20.3%

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

   7. Simplified 20.3%

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

   8. Taylor expanded in x around inf 37.1%

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

      1. unpow2 37.1%

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

   10. Simplified 37.1%

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

   11. Taylor expanded in t around -inf 82.0%

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

       1. mul-1-neg 82.0%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 82.0%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -6.60000000000000004e-104 < t < -1.0200000000000001e-196 or -1.8e-302 < t < 6.00000000000000037e-140

   1. Initial program 10.6%

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

      1. associate-*l/ 10.6%

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

   3. Simplified 10.6%

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

      1. *-commutative 10.6%

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

      2. clear-num 10.6%

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

      3. un-div-inv 10.6%

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

      4. sqrt-undiv 10.6%

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

      5. metadata-eval 10.6%

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

      6. sub-neg 10.6%

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

      7. associate-*l/ 13.2%

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

      8. sub-neg 13.2%

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

      9. metadata-eval 13.2%

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

   5. Applied egg-rr 13.2%

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

      1. +-commutative 13.2%

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

      2. *-commutative 13.2%

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

      3. +-commutative 13.2%

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

      4. +-commutative 13.2%

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

   7. Simplified 13.2%

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

   8. Taylor expanded in x around inf 67.5%

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

      1. cancel-sign-sub-inv 67.5%

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

      2. fma-def 67.5%

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

      3. unpow2 67.5%

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

      4. fma-def 67.5%

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

      5. unpow2 67.5%

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

      6. unpow2 67.5%

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

      7. metadata-eval 67.5%

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

      8. unpow2 67.5%

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

      9. unpow2 67.5%

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

      10. fma-udef 67.5%

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

      11. *-lft-identity 67.5%

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

      12. fma-udef 67.5%

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

      13. unpow2 67.5%

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

      14. unpow2 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in t around 0 47.9%

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

       1. associate-*l* 47.9%

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

       2. *-commutative 47.9%

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

       3. associate-*l* 48.0%

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

   13. Simplified 48.0%

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

       1. expm1-log1p-u 39.1%

          \[\leadsto \frac{t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{1}{x}}\right)\right)\right)\right)}} \]

       2. expm1-udef 24.9%

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

       3. associate-*r* 24.9%

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

       4. sqrt-unprod 24.9%

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

       5. metadata-eval 24.9%

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

       6. metadata-eval 24.9%

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

       7. *-un-lft-identity 24.9%

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

       8. add-sqr-sqrt 24.3%

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

       9. sqrt-prod 35.6%

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

       10. sqrt-prod 35.9%

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

       11. div-inv 35.9%

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

       12. sqrt-div 35.6%

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

       13. sqrt-prod 24.3%

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

       14. add-sqr-sqrt 24.9%

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

   15. Applied egg-rr 24.9%

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

       1. expm1-def 39.2%

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

       2. expm1-log1p 48.2%

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

   17. Simplified 48.2%

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

   if 6.00000000000000037e-140 < t

   1. Initial program 38.4%

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

      1. associate-*l/ 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in t around inf 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-*r/ 49.9%

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

      3. sub-neg 49.9%

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

      4. metadata-eval 49.9%

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

      5. unpow2 49.9%

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

      6. +-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in x around inf 82.3%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   8. Step-by-step derivation

      1. associate--l+ 82.3%

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

      2. associate-*r/ 82.3%

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

      3. metadata-eval 82.3%

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

      4. unpow2 82.3%

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

   9. Simplified 82.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-196}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-140}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6: 76.4% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (if (<= t -2e-310) (/ t (- t)) (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))

C

double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-310) {
		tmp = t / -t;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-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 = t / -t
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-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 = t / -t;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = t / -t
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(-t));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2e-310)
		tmp = t / -t;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(t / (-t)), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 30.5%

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

      1. associate-*l/ 30.5%

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

   3. Simplified 30.5%

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

      1. *-commutative 30.5%

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

      2. clear-num 30.5%

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

      3. un-div-inv 30.5%

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

      4. sqrt-undiv 30.6%

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

      5. metadata-eval 30.6%

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

      6. sub-neg 30.6%

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

      7. associate-*l/ 19.8%

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

      8. sub-neg 19.8%

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

      9. metadata-eval 19.8%

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

   5. Applied egg-rr 19.8%

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

      1. +-commutative 19.8%

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

      2. *-commutative 19.8%

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

      3. +-commutative 19.8%

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

      4. +-commutative 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in x around inf 35.5%

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

      1. unpow2 35.5%

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

   10. Simplified 35.5%

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

   11. Taylor expanded in t around -inf 74.3%

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

       1. mul-1-neg 74.3%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 74.3%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 32.4%

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

      1. associate-*l/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in t around inf 30.8%

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

      1. +-commutative 30.8%

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

      2. associate-*r/ 41.8%

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

      3. sub-neg 41.8%

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

      4. metadata-eval 41.8%

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

      5. unpow2 41.8%

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

      6. +-commutative 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in x around inf 71.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   8. Step-by-step derivation

      1. associate--l+ 71.6%

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

      2. associate-*r/ 71.6%

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

      3. metadata-eval 71.6%

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

      4. unpow2 71.6%

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

   9. Simplified 71.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t}{-t}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7: 76.3% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-310)
		tmp = Float64(t / Float64(-t));
	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 = t / -t;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, l_, t_] := If[LessEqual[t, -2e-310], N[(t / (-t)), $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 30.5%

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

      1. associate-*l/ 30.5%

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

   3. Simplified 30.5%

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

      1. *-commutative 30.5%

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

      2. clear-num 30.5%

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

      3. un-div-inv 30.5%

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

      4. sqrt-undiv 30.6%

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

      5. metadata-eval 30.6%

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

      6. sub-neg 30.6%

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

      7. associate-*l/ 19.8%

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

      8. sub-neg 19.8%

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

      9. metadata-eval 19.8%

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

   5. Applied egg-rr 19.8%

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

      1. +-commutative 19.8%

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

      2. *-commutative 19.8%

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

      3. +-commutative 19.8%

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

      4. +-commutative 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in x around inf 35.5%

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

      1. unpow2 35.5%

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

   10. Simplified 35.5%

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

   11. Taylor expanded in t around -inf 74.3%

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

       1. mul-1-neg 74.3%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 74.3%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 32.4%

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

      1. associate-*l/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in t around inf 30.8%

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

      1. +-commutative 30.8%

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

      2. associate-*r/ 41.8%

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

      3. sub-neg 41.8%

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

      4. metadata-eval 41.8%

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

      5. unpow2 41.8%

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

      6. +-commutative 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in x around inf 71.4%

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

4. Final simplification 72.9%

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

Alternative 8: 75.9% accurate, 37.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 30.5%

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

      1. associate-*l/ 30.5%

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

   3. Simplified 30.5%

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

      1. *-commutative 30.5%

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

      2. clear-num 30.5%

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

      3. un-div-inv 30.5%

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

      4. sqrt-undiv 30.6%

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

      5. metadata-eval 30.6%

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

      6. sub-neg 30.6%

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

      7. associate-*l/ 19.8%

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

      8. sub-neg 19.8%

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

      9. metadata-eval 19.8%

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

   5. Applied egg-rr 19.8%

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

      1. +-commutative 19.8%

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

      2. *-commutative 19.8%

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

      3. +-commutative 19.8%

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

      4. +-commutative 19.8%

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

   7. Simplified 19.8%

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

   8. Taylor expanded in x around inf 35.5%

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

      1. unpow2 35.5%

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

   10. Simplified 35.5%

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

   11. Taylor expanded in t around -inf 74.3%

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

       1. mul-1-neg 74.3%

          \[\leadsto \frac{t}{\color{blue}{-t}} \]

   13. Simplified 74.3%

       \[\leadsto \frac{t}{\color{blue}{-t}} \]

   if -1.999999999999994e-310 < t

   1. Initial program 32.4%

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

      1. associate-*l/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in t around inf 30.8%

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

      1. +-commutative 30.8%

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

      2. associate-*r/ 41.8%

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

      3. sub-neg 41.8%

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

      4. metadata-eval 41.8%

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

      5. unpow2 41.8%

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

      6. +-commutative 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in x around inf 70.8%

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

4. Final simplification 72.7%

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

Alternative 9: 38.7% 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 31.4%

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

   1. associate-*l/ 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in t around inf 26.7%

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

   1. +-commutative 26.7%

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

   2. associate-*r/ 38.8%

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

   3. sub-neg 38.8%

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

   4. metadata-eval 38.8%

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

   5. unpow2 38.8%

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

   6. +-commutative 38.8%

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

6. Simplified 38.8%

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

7. Taylor expanded in x around inf 35.0%

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

8. Final simplification 35.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023292 
(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)))))