Toniolo and Linder, Equation (7)

Percentage Accurate: 32.1% → 83.0%

Time: 10.4s

Alternatives: 9

Speedup: 85.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, l, t)
use fmin_fmax_functions
    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: 32.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, l, t)
use fmin_fmax_functions
    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: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \frac{t\_2}{x}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{t\_m \cdot t\_m}{x}\right), \frac{\ell \cdot \ell}{x}\right) + t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{t\_2 \cdot 2}{x}}{\sqrt{2}}, \frac{0.5}{t\_m}, \frac{\frac{t\_2}{t\_m}}{x \cdot \sqrt{2}}\right), 0.5, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\left(t\_3 + \mathsf{fma}\left(t\_m \cdot t\_m, 2, \mathsf{fma}\left(\ell, \ell, t\_2\right)\right)\right) + t\_3}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l)))
        (t_3 (/ t_2 x))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.12e-270)
      (/
       t_4
       (sqrt (+ (fma 2.0 (fma t_m t_m (/ (* t_m t_m) x)) (/ (* l l) x)) t_3)))
      (if (<= t_m 1.45e-162)
        (/
         t_4
         (fma
          (fma
           (/ (/ (* t_2 2.0) x) (sqrt 2.0))
           (/ 0.5 t_m)
           (/ (/ t_2 t_m) (* x (sqrt 2.0))))
          0.5
          t_4))
        (if (<= t_m 8.5e+65)
          (/
           t_4
           (sqrt
            (fma
             (* t_m t_m)
             2.0
             (/ (+ (+ t_3 (fma (* t_m t_m) 2.0 (fma l l t_2))) t_3) x))))
          1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = t_2 / x;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.12e-270) {
		tmp = t_4 / sqrt((fma(2.0, fma(t_m, t_m, ((t_m * t_m) / x)), ((l * l) / x)) + t_3));
	} else if (t_m <= 1.45e-162) {
		tmp = t_4 / fma(fma((((t_2 * 2.0) / x) / sqrt(2.0)), (0.5 / t_m), ((t_2 / t_m) / (x * sqrt(2.0)))), 0.5, t_4);
	} else if (t_m <= 8.5e+65) {
		tmp = t_4 / sqrt(fma((t_m * t_m), 2.0, (((t_3 + fma((t_m * t_m), 2.0, fma(l, l, t_2))) + t_3) / x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(t_2 / x)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.12e-270)
		tmp = Float64(t_4 / sqrt(Float64(fma(2.0, fma(t_m, t_m, Float64(Float64(t_m * t_m) / x)), Float64(Float64(l * l) / x)) + t_3)));
	elseif (t_m <= 1.45e-162)
		tmp = Float64(t_4 / fma(fma(Float64(Float64(Float64(t_2 * 2.0) / x) / sqrt(2.0)), Float64(0.5 / t_m), Float64(Float64(t_2 / t_m) / Float64(x * sqrt(2.0)))), 0.5, t_4));
	elseif (t_m <= 8.5e+65)
		tmp = Float64(t_4 / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(Float64(Float64(t_3 + fma(Float64(t_m * t_m), 2.0, fma(l, l, t_2))) + t_3) / x))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.12e-270], N[(t$95$4 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e-162], N[(t$95$4 / N[(N[(N[(N[(N[(t$95$2 * 2.0), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 / t$95$m), $MachinePrecision] + N[(N[(t$95$2 / t$95$m), $MachinePrecision] / N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+65], N[(t$95$4 / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(t$95$3 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.1199999999999999e-270

   1. Initial program 29.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 55.9%

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

   if 1.1199999999999999e-270 < t < 1.4500000000000001e-162

   1. Initial program 2.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 80.2%

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

   if 1.4500000000000001e-162 < t < 8.50000000000000075e65

   1. Initial program 55.2%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 86.7%

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

   if 8.50000000000000075e65 < t

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

      3. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 98.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right), \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right) \cdot 2}{x}}{\sqrt{2}}, \frac{0.5}{t}, \frac{\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{t}}{x \cdot \sqrt{2}}\right), 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, \frac{\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(t \cdot t, 2, \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)\right)\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)\\ t_3 := \frac{t\_2}{x}\\ t_4 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{t\_m \cdot t\_m}{x}\right), \frac{\ell \cdot \ell}{x}\right) + t\_3}}\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_4\right)}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \frac{\left(t\_3 + \mathsf{fma}\left(t\_m \cdot t\_m, 2, \mathsf{fma}\left(\ell, \ell, t\_2\right)\right)\right) + t\_3}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (fma (* t_m t_m) 2.0 (* l l)))
        (t_3 (/ t_2 x))
        (t_4 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.12e-270)
      (/
       t_4
       (sqrt (+ (fma 2.0 (fma t_m t_m (/ (* t_m t_m) x)) (/ (* l l) x)) t_3)))
      (if (<= t_m 1.45e-162)
        (/ t_4 (fma (/ (* (* l l) 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_4))
        (if (<= t_m 8.5e+65)
          (/
           t_4
           (sqrt
            (fma
             (* t_m t_m)
             2.0
             (/ (+ (+ t_3 (fma (* t_m t_m) 2.0 (fma l l t_2))) t_3) x))))
          1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = fma((t_m * t_m), 2.0, (l * l));
	double t_3 = t_2 / x;
	double t_4 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.12e-270) {
		tmp = t_4 / sqrt((fma(2.0, fma(t_m, t_m, ((t_m * t_m) / x)), ((l * l) / x)) + t_3));
	} else if (t_m <= 1.45e-162) {
		tmp = t_4 / fma((((l * l) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_4);
	} else if (t_m <= 8.5e+65) {
		tmp = t_4 / sqrt(fma((t_m * t_m), 2.0, (((t_3 + fma((t_m * t_m), 2.0, fma(l, l, t_2))) + t_3) / x)));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = fma(Float64(t_m * t_m), 2.0, Float64(l * l))
	t_3 = Float64(t_2 / x)
	t_4 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.12e-270)
		tmp = Float64(t_4 / sqrt(Float64(fma(2.0, fma(t_m, t_m, Float64(Float64(t_m * t_m) / x)), Float64(Float64(l * l) / x)) + t_3)));
	elseif (t_m <= 1.45e-162)
		tmp = Float64(t_4 / fma(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_4));
	elseif (t_m <= 8.5e+65)
		tmp = Float64(t_4 / sqrt(fma(Float64(t_m * t_m), 2.0, Float64(Float64(Float64(t_3 + fma(Float64(t_m * t_m), 2.0, fma(l, l, t_2))) + t_3) / x))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.12e-270], N[(t$95$4 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e-162], N[(t$95$4 / N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+65], N[(t$95$4 / N[Sqrt[N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(N[(N[(t$95$3 + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.1199999999999999e-270

   1. Initial program 29.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 55.9%

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

   if 1.1199999999999999e-270 < t < 1.4500000000000001e-162

   1. Initial program 2.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 80.2%

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

   if 1.4500000000000001e-162 < t < 8.50000000000000075e65

   1. Initial program 55.2%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 86.7%

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

   if 8.50000000000000075e65 < t

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

      3. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 98.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right), \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(t \cdot t, 2, \frac{\left(\frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x} + \mathsf{fma}\left(t \cdot t, 2, \mathsf{fma}\left(\ell, \ell, \mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)\right)\right)\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t_3 := \frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t\_m, t\_m, \frac{t\_m \cdot t\_m}{x}\right), \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right)}{x}}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m))
        (t_3
         (/
          t_2
          (sqrt
           (+
            (fma 2.0 (fma t_m t_m (/ (* t_m t_m) x)) (/ (* l l) x))
            (/ (fma (* t_m t_m) 2.0 (* l l)) x))))))
   (*
    t_s
    (if (<= t_m 1.12e-270)
      t_3
      (if (<= t_m 1.45e-162)
        (/ t_2 (fma (/ (* (* l l) 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
        (if (<= t_m 8.5e+65) t_3 1.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double t_3 = t_2 / sqrt((fma(2.0, fma(t_m, t_m, ((t_m * t_m) / x)), ((l * l) / x)) + (fma((t_m * t_m), 2.0, (l * l)) / x)));
	double tmp;
	if (t_m <= 1.12e-270) {
		tmp = t_3;
	} else if (t_m <= 1.45e-162) {
		tmp = t_2 / fma((((l * l) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else if (t_m <= 8.5e+65) {
		tmp = t_3;
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	t_3 = Float64(t_2 / sqrt(Float64(fma(2.0, fma(t_m, t_m, Float64(Float64(t_m * t_m) / x)), Float64(Float64(l * l) / x)) + Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) / x))))
	tmp = 0.0
	if (t_m <= 1.12e-270)
		tmp = t_3;
	elseif (t_m <= 1.45e-162)
		tmp = Float64(t_2 / fma(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	elseif (t_m <= 8.5e+65)
		tmp = t_3;
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(t$95$m * t$95$m + N[(N[(t$95$m * t$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.12e-270], t$95$3, If[LessEqual[t$95$m, 1.45e-162], N[(t$95$2 / N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+65], t$95$3, 1.0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.1199999999999999e-270 or 1.4500000000000001e-162 < t < 8.50000000000000075e65

   1. Initial program 37.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 29.5%

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

   6. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

   8. Applied rewrites 65.5%

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

   if 1.1199999999999999e-270 < t < 1.4500000000000001e-162

   1. Initial program 2.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 80.2%

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

   if 8.50000000000000075e65 < t

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

      3. lower-sqrt.f64 96.5

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

   5. Applied rewrites 96.5%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right), \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t}, 0.5, \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right), \frac{\ell \cdot \ell}{x}\right) + \frac{\mathsf{fma}\left(t \cdot t, 2, \ell \cdot \ell\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\sqrt{\frac{x - 1}{1 + x}} \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\left(\frac{\ell}{\left(x \cdot t\_m\right) \cdot t\_m} \cdot \frac{\ell}{\sqrt{2}} + \sqrt{2}\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-324)
    (* (sqrt (/ (- x 1.0) (+ 1.0 x))) (* (sqrt 0.5) (sqrt 2.0)))
    (/
     (* (sqrt 2.0) t_m)
     (* (+ (* (/ l (* (* x t_m) t_m)) (/ l (sqrt 2.0))) (sqrt 2.0)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if ((l * l) <= 5e-324) {
		tmp = sqrt(((x - 1.0) / (1.0 + x))) * (sqrt(0.5) * sqrt(2.0));
	} else {
		tmp = (sqrt(2.0) * t_m) / ((((l / ((x * t_m) * t_m)) * (l / sqrt(2.0))) + sqrt(2.0)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l * l) <= 5d-324) then
        tmp = sqrt(((x - 1.0d0) / (1.0d0 + x))) * (sqrt(0.5d0) * sqrt(2.0d0))
    else
        tmp = (sqrt(2.0d0) * t_m) / ((((l / ((x * t_m) * t_m)) * (l / sqrt(2.0d0))) + sqrt(2.0d0)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if ((l * l) <= 5e-324) {
		tmp = Math.sqrt(((x - 1.0) / (1.0 + x))) * (Math.sqrt(0.5) * Math.sqrt(2.0));
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / ((((l / ((x * t_m) * t_m)) * (l / Math.sqrt(2.0))) + Math.sqrt(2.0)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if (l * l) <= 5e-324:
		tmp = math.sqrt(((x - 1.0) / (1.0 + x))) * (math.sqrt(0.5) * math.sqrt(2.0))
	else:
		tmp = (math.sqrt(2.0) * t_m) / ((((l / ((x * t_m) * t_m)) * (l / math.sqrt(2.0))) + math.sqrt(2.0)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e-324)
		tmp = Float64(sqrt(Float64(Float64(x - 1.0) / Float64(1.0 + x))) * Float64(sqrt(0.5) * sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(Float64(Float64(Float64(l / Float64(Float64(x * t_m) * t_m)) * Float64(l / sqrt(2.0))) + sqrt(2.0)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if ((l * l) <= 5e-324)
		tmp = sqrt(((x - 1.0) / (1.0 + x))) * (sqrt(0.5) * sqrt(2.0));
	else
		tmp = (sqrt(2.0) * t_m) / ((((l / ((x * t_m) * t_m)) * (l / sqrt(2.0))) + sqrt(2.0)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-324], N[(N[Sqrt[N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[(N[(N[(l / N[(N[(x * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-324

   1. Initial program 46.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if 4.94066e-324 < (*.f64 l l)

   1. Initial program 27.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in l around inf

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

   11. Applied rewrites 40.3%

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

   13. Applied rewrites 41.8%

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

Alternative 5: 79.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * ((sqrt(2.0) * t_m) / (((fma((l / t_m), (l / t_m), 2.0) / (sqrt(2.0) * x)) + sqrt(2.0)) * t_m));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 33.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 47.2%

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

Alternative 6: 77.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	return t_s * ((sqrt(2.0) * t_m) / ((((((l / t_m) / t_m) * l) / (x * sqrt(2.0))) + sqrt(2.0)) * t_m));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * ((sqrt(2.0d0) * t_m) / ((((((l / t_m) / t_m) * l) / (x * sqrt(2.0d0))) + sqrt(2.0d0)) * t_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 33.9%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 47.2%

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

9. Taylor expanded in l around inf

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

11. Applied rewrites 41.9%

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

13. Applied rewrites 46.7%

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

Alternative 7: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{2} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-34}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\sqrt{2} \cdot x\right) \cdot t\_m}, 0.5, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (let* ((t_2 (* (sqrt 2.0) t_m)))
   (*
    t_s
    (if (<= t_m 1.3e-34)
      (/ t_2 (fma (/ (* (* l l) 2.0) (* (* (sqrt 2.0) x) t_m)) 0.5 t_2))
      1.0))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double t_2 = sqrt(2.0) * t_m;
	double tmp;
	if (t_m <= 1.3e-34) {
		tmp = t_2 / fma((((l * l) * 2.0) / ((sqrt(2.0) * x) * t_m)), 0.5, t_2);
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	t_2 = Float64(sqrt(2.0) * t_m)
	tmp = 0.0
	if (t_m <= 1.3e-34)
		tmp = Float64(t_2 / fma(Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(sqrt(2.0) * x) * t_m)), 0.5, t_2));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-34], N[(t$95$2 / N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$2), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e-34

   1. Initial program 25.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 23.9%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 24.0%

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

   if 1.3e-34 < t

   1. Initial program 47.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

      3. lower-sqrt.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 96.2%

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

Alternative 8: 74.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-267}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\ell \cdot \frac{\ell}{x \cdot \left(t\_m \cdot \sqrt{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-267)
    (/ (* (sqrt 2.0) t_m) (* l (/ l (* x (* t_m (sqrt 2.0))))))
    1.0)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.95e-267) {
		tmp = (sqrt(2.0) * t_m) / (l * (l / (x * (t_m * sqrt(2.0)))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 1.95d-267) then
        tmp = (sqrt(2.0d0) * t_m) / (l * (l / (x * (t_m * sqrt(2.0d0)))))
    else
        tmp = 1.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l, double t_m) {
	double tmp;
	if (t_m <= 1.95e-267) {
		tmp = (Math.sqrt(2.0) * t_m) / (l * (l / (x * (t_m * Math.sqrt(2.0)))));
	} else {
		tmp = 1.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l, t_m):
	tmp = 0
	if t_m <= 1.95e-267:
		tmp = (math.sqrt(2.0) * t_m) / (l * (l / (x * (t_m * math.sqrt(2.0)))))
	else:
		tmp = 1.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l, t_m)
	tmp = 0.0
	if (t_m <= 1.95e-267)
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(l * Float64(l / Float64(x * Float64(t_m * sqrt(2.0))))));
	else
		tmp = 1.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l, t_m)
	tmp = 0.0;
	if (t_m <= 1.95e-267)
		tmp = (sqrt(2.0) * t_m) / (l * (l / (x * (t_m * sqrt(2.0)))));
	else
		tmp = 1.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-267], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * N[(l / N[(x * N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999988e-267

   1. Initial program 29.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 10.1%

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

   10. Applied rewrites 10.1%

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

   if 1.94999999999999988e-267 < t

   1. Initial program 35.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

      3. lower-sqrt.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 81.0%

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

Alternative 9: 74.1% accurate, 85.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, x, l, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2} \]

   3. lower-sqrt.f64 40.9

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

5. Applied rewrites 40.9%

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

7. Applied rewrites 41.5%

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

Reproduce

herbie shell --seed 2025015 
(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)))))