Toniolo and Linder, Equation (7)

Percentage Accurate: 33.6% → 44.4%

Time: 10.0s

Alternatives: 7

Speedup: 1.2×

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: 33.6% 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: 44.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1 + x}{x - 1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.55 \cdot 10^{-158}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, 2 \cdot \left(t\_m \cdot t\_m\right), \ell \cdot \mathsf{fma}\left(\ell, t\_2, -\ell\right)\right)}} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 (/ (+ 1.0 x) (- x 1.0))))
   (*
    t_s
    (if (<= t_m 2.55e-158)
      (*
       t_m
       (/
        (sqrt 2.0)
        (sqrt (fma (- l) l (* (fma (* t_m t_m) 2.0 (* l l)) t_2)))))
      (*
       (sqrt
        (/
         2.0
         (fma
          (/ (+ x 1.0) (- x 1.0))
          (* 2.0 (* t_m t_m))
          (* l (fma l t_2 (- l))))))
       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 t_2 = (1.0 + x) / (x - 1.0);
	double tmp;
	if (t_m <= 2.55e-158) {
		tmp = t_m * (sqrt(2.0) / sqrt(fma(-l, l, (fma((t_m * t_m), 2.0, (l * l)) * t_2))));
	} else {
		tmp = sqrt((2.0 / fma(((x + 1.0) / (x - 1.0)), (2.0 * (t_m * t_m)), (l * fma(l, t_2, -l))))) * 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)
	t_2 = Float64(Float64(1.0 + x) / Float64(x - 1.0))
	tmp = 0.0
	if (t_m <= 2.55e-158)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(fma(Float64(-l), l, Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * t_2)))));
	else
		tmp = Float64(sqrt(Float64(2.0 / fma(Float64(Float64(x + 1.0) / Float64(x - 1.0)), Float64(2.0 * Float64(t_m * t_m)), Float64(l * fma(l, t_2, Float64(-l)))))) * t_m);
	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[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.55e-158], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[((-l) * l + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 / N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l * t$95$2 + (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5500000000000002e-158

   1. Initial program 33.0%

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

   4. Applied rewrites 40.1%

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

   if 2.5500000000000002e-158 < t

   1. Initial program 40.5%

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

   4. Applied rewrites 44.0%

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

   6. Applied rewrites 40.5%

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

   8. Applied rewrites 46.7%

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

   10. Applied rewrites 51.1%

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

Alternative 2: 43.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.62 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, 2 \cdot \left(t\_m \cdot t\_m\right), \ell \cdot \left(\frac{\left(1 + x\right) \cdot \ell}{x - 1} - \ell\right)\right)}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(1 + x, \left(t\_m \cdot 2\right) \cdot \frac{t\_m}{x - 1}, \ell \cdot \mathsf{fma}\left(\ell, \frac{1 + x}{x - 1}, -\ell\right)\right)}} \cdot t\_m\\ \end{array} \end{array} \]

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 (<= t_m 1.62e-15)
    (*
     (sqrt
      (/
       2.0
       (fma
        (/ (+ x 1.0) (- x 1.0))
        (* 2.0 (* t_m t_m))
        (* l (- (/ (* (+ 1.0 x) l) (- x 1.0)) l)))))
     t_m)
    (*
     (sqrt
      (/
       2.0
       (fma
        (+ 1.0 x)
        (* (* t_m 2.0) (/ t_m (- x 1.0)))
        (* l (fma l (/ (+ 1.0 x) (- x 1.0)) (- l))))))
     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 (t_m <= 1.62e-15) {
		tmp = sqrt((2.0 / fma(((x + 1.0) / (x - 1.0)), (2.0 * (t_m * t_m)), (l * ((((1.0 + x) * l) / (x - 1.0)) - l))))) * t_m;
	} else {
		tmp = sqrt((2.0 / fma((1.0 + x), ((t_m * 2.0) * (t_m / (x - 1.0))), (l * fma(l, ((1.0 + x) / (x - 1.0)), -l))))) * 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 (t_m <= 1.62e-15)
		tmp = Float64(sqrt(Float64(2.0 / fma(Float64(Float64(x + 1.0) / Float64(x - 1.0)), Float64(2.0 * Float64(t_m * t_m)), Float64(l * Float64(Float64(Float64(Float64(1.0 + x) * l) / Float64(x - 1.0)) - l))))) * t_m);
	else
		tmp = Float64(sqrt(Float64(2.0 / fma(Float64(1.0 + x), Float64(Float64(t_m * 2.0) * Float64(t_m / Float64(x - 1.0))), Float64(l * fma(l, Float64(Float64(1.0 + x) / Float64(x - 1.0)), Float64(-l)))))) * t_m);
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.62e-15], N[(N[Sqrt[N[(2.0 / N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l * N[(N[(N[(N[(1.0 + x), $MachinePrecision] * l), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[Sqrt[N[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(t$95$m / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l * N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] + (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.62000000000000009e-15

   1. Initial program 36.3%

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

   4. Applied rewrites 43.9%

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

   6. Applied rewrites 35.9%

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

   8. Applied rewrites 42.9%

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

   10. Applied rewrites 45.4%

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

   if 1.62000000000000009e-15 < t

   1. Initial program 36.3%

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

   4. Applied rewrites 36.8%

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

   6. Applied rewrites 36.3%

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

   8. Applied rewrites 40.2%

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

   10. Applied rewrites 45.0%

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

Alternative 3: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1 + x}{x - 1}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-35}:\\ \;\;\;\;t\_m \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(-\ell, \ell, \mathsf{fma}\left(t\_m \cdot t\_m, 2, \ell \cdot \ell\right) \cdot t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(1 + x, \left(t\_m \cdot 2\right) \cdot \frac{t\_m}{x - 1}, \ell \cdot \mathsf{fma}\left(\ell, t\_2, -\ell\right)\right)}} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 (/ (+ 1.0 x) (- x 1.0))))
   (*
    t_s
    (if (<= t_m 1.1e-35)
      (*
       t_m
       (/
        (sqrt 2.0)
        (sqrt (fma (- l) l (* (fma (* t_m t_m) 2.0 (* l l)) t_2)))))
      (*
       (sqrt
        (/
         2.0
         (fma
          (+ 1.0 x)
          (* (* t_m 2.0) (/ t_m (- x 1.0)))
          (* l (fma l t_2 (- l))))))
       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 t_2 = (1.0 + x) / (x - 1.0);
	double tmp;
	if (t_m <= 1.1e-35) {
		tmp = t_m * (sqrt(2.0) / sqrt(fma(-l, l, (fma((t_m * t_m), 2.0, (l * l)) * t_2))));
	} else {
		tmp = sqrt((2.0 / fma((1.0 + x), ((t_m * 2.0) * (t_m / (x - 1.0))), (l * fma(l, t_2, -l))))) * 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)
	t_2 = Float64(Float64(1.0 + x) / Float64(x - 1.0))
	tmp = 0.0
	if (t_m <= 1.1e-35)
		tmp = Float64(t_m * Float64(sqrt(2.0) / sqrt(fma(Float64(-l), l, Float64(fma(Float64(t_m * t_m), 2.0, Float64(l * l)) * t_2)))));
	else
		tmp = Float64(sqrt(Float64(2.0 / fma(Float64(1.0 + x), Float64(Float64(t_m * 2.0) * Float64(t_m / Float64(x - 1.0))), Float64(l * fma(l, t_2, Float64(-l)))))) * t_m);
	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[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.1e-35], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[((-l) * l + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(t$95$m / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l * t$95$2 + (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.09999999999999997e-35

   1. Initial program 35.9%

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

   4. Applied rewrites 43.6%

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

   if 1.09999999999999997e-35 < t

   1. Initial program 37.3%

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

   4. Applied rewrites 38.0%

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

   6. Applied rewrites 37.3%

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

   8. Applied rewrites 42.1%

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

   10. Applied rewrites 46.7%

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

Alternative 4: 38.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 41.8%

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

Alternative 5: 35.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(2, t\_m \cdot t\_m, \ell \cdot \ell\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.65 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\frac{2}{\frac{x + 1}{x - 1} \cdot t\_2 - \ell \cdot \ell}} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x + 1, x, 1\right), -1 \cdot t\_2, \left(-\ell\right) \cdot \ell\right)}} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 2.0 (* t_m t_m) (* l l))))
   (*
    t_s
    (if (<= l 1.65e+109)
      (* (sqrt (/ 2.0 (- (* (/ (+ x 1.0) (- x 1.0)) t_2) (* l l)))) t_m)
      (*
       (sqrt (/ 2.0 (fma (fma (+ x 1.0) x 1.0) (* -1.0 t_2) (* (- l) l))))
       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 t_2 = fma(2.0, (t_m * t_m), (l * l));
	double tmp;
	if (l <= 1.65e+109) {
		tmp = sqrt((2.0 / ((((x + 1.0) / (x - 1.0)) * t_2) - (l * l)))) * t_m;
	} else {
		tmp = sqrt((2.0 / fma(fma((x + 1.0), x, 1.0), (-1.0 * t_2), (-l * l)))) * 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)
	t_2 = fma(2.0, Float64(t_m * t_m), Float64(l * l))
	tmp = 0.0
	if (l <= 1.65e+109)
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * t_2) - Float64(l * l)))) * t_m);
	else
		tmp = Float64(sqrt(Float64(2.0 / fma(fma(Float64(x + 1.0), x, 1.0), Float64(-1.0 * t_2), Float64(Float64(-l) * l)))) * t_m);
	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[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 1.65e+109], N[(N[Sqrt[N[(2.0 / N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[Sqrt[N[(2.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(-1.0 * t$95$2), $MachinePrecision] + N[((-l) * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.6499999999999999e109

   1. Initial program 41.7%

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

   4. Applied rewrites 44.2%

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

   6. Applied rewrites 41.3%

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

   8. Applied rewrites 41.4%

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

   if 1.6499999999999999e109 < l

   1. Initial program 0.9%

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

   4. Applied rewrites 26.5%

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

   6. Applied rewrites 0.9%

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

   8. Applied rewrites 0.9%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 26.6%

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

Alternative 6: 38.1% accurate, 1.2× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (*
   (sqrt
    (/
     2.0
     (fma (- l) l (* (/ (+ x 1.0) (- x 1.0)) (fma 2.0 (* t_m t_m) (* l l))))))
   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 / fma(-l, l, (((x + 1.0) / (x - 1.0)) * fma(2.0, (t_m * t_m), (l * l)))))) * 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(sqrt(Float64(2.0 / fma(Float64(-l), l, Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * fma(2.0, Float64(t_m * t_m), Float64(l * l)))))) * 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[Sqrt[N[(2.0 / N[((-l) * l + N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 41.8%

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

6. Applied rewrites 36.0%

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

8. Applied rewrites 41.5%

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

Alternative 7: 6.7% accurate, 1.2× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l t_m)
 :precision binary64
 (*
  t_s
  (*
   (sqrt
    (/
     2.0
     (fma
      (fma (+ x 1.0) x 1.0)
      (* -1.0 (fma 2.0 (* t_m t_m) (* l l)))
      (* (- l) l))))
   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 / fma(fma((x + 1.0), x, 1.0), (-1.0 * fma(2.0, (t_m * t_m), (l * l))), (-l * l)))) * 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(sqrt(Float64(2.0 / fma(fma(Float64(x + 1.0), x, 1.0), Float64(-1.0 * fma(2.0, Float64(t_m * t_m), Float64(l * l))), Float64(Float64(-l) * l)))) * 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[Sqrt[N[(2.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(-1.0 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-l) * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

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

4. Applied rewrites 41.8%

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

6. Applied rewrites 36.0%

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

8. Applied rewrites 11.4%

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

9. Taylor expanded in x around 0

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

11. Simplified 7.3%

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

Reproduce

herbie shell --seed 2025017 -o generate:proofs
(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)))))