Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.7%

Time: 9.3s

Alternatives: 12

Speedup: 0.5×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

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(p, x)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

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(p, x)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \left(4 \cdot p\_m\right) \cdot p\_m\\ t_1 := \frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{t\_0 + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p\_m \cdot 4\right) \cdot p\_m\right)}}, \frac{{t\_1}^{2} - 1}{{t\_1}^{3} - -1} \cdot \mathsf{fma}\left(t\_1 - 1, t\_1, 1\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* (* 4.0 p_m) p_m)) (t_1 (/ x (sqrt (fma x x t_0)))))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ t_0 (* x x))))))) 0.0)
     (/ (- p_m) x)
     (sqrt
      (/
       (fma (pow (/ x (sqrt (fma (* p_m p_m) 4.0 (* x x)))) 3.0) 0.5 0.5)
       (fma
        (/ x (sqrt (fma x x (* (* p_m 4.0) p_m))))
        (*
         (/ (- (pow t_1 2.0) 1.0) (- (pow t_1 3.0) -1.0))
         (fma (- t_1 1.0) t_1 1.0))
        1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = (4.0 * p_m) * p_m;
	double t_1 = x / sqrt(fma(x, x, t_0));
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((t_0 + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((fma(pow((x / sqrt(fma((p_m * p_m), 4.0, (x * x)))), 3.0), 0.5, 0.5) / fma((x / sqrt(fma(x, x, ((p_m * 4.0) * p_m)))), (((pow(t_1, 2.0) - 1.0) / (pow(t_1, 3.0) - -1.0)) * fma((t_1 - 1.0), t_1, 1.0)), 1.0)));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(4.0 * p_m) * p_m)
	t_1 = Float64(x / sqrt(fma(x, x, t_0)))
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(t_0 + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(fma((Float64(x / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))) ^ 3.0), 0.5, 0.5) / fma(Float64(x / sqrt(fma(x, x, Float64(Float64(p_m * 4.0) * p_m)))), Float64(Float64(Float64((t_1 ^ 2.0) - 1.0) / Float64((t_1 ^ 3.0) - -1.0)) * fma(Float64(t_1 - 1.0), t_1, 1.0)), 1.0)));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(N[Power[N[(x / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] / N[(N[(x / N[Sqrt[N[(x * x + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Power[t$95$1, 3.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 - 1.0), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{2} \cdot \color{blue}{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\frac{\color{blue}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      6. lower-fma.f64 99.7

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3}, 0.5, 0.5\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(p \cdot 4\right) \cdot p}}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(p \cdot 4\right) \cdot p}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot 4\right) \cdot p + x \cdot x}}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot 4\right) \cdot p} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{p \cdot \left(p \cdot 4\right)} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{p \cdot \color{blue}{\left(p \cdot 4\right)} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot p\right) \cdot 4} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot p\right)} \cdot 4 + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      15. lower-fma.f64 99.7

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}\right)}^{2} - 1}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}\right)}^{3} - -1} \cdot \mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}} - 1, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 1\right)}, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}\right)}^{2} - 1}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}\right)}^{3} - -1} \cdot \mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}} - 1, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 1\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p\_m \cdot 4\right) \cdot p\_m\right)}}\\ t_1 := \left(4 \cdot p\_m\right) \cdot p\_m\\ t_2 := \frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_1\right)}}\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{t\_1 + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left({t\_0}^{3} - -1\right) \cdot 0.5}{\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(t\_2, t\_2, -1\right)}{\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}} - -1}, 1\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (fma x x (* (* p_m 4.0) p_m)))))
        (t_1 (* (* 4.0 p_m) p_m))
        (t_2 (/ x (sqrt (fma x x t_1)))))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ t_1 (* x x))))))) 0.0)
     (/ (- p_m) x)
     (sqrt
      (/
       (* (- (pow t_0 3.0) -1.0) 0.5)
       (fma
        t_0
        (/
         (fma t_2 t_2 -1.0)
         (- (/ x (sqrt (fma (* p_m p_m) 4.0 (* x x)))) -1.0))
        1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(fma(x, x, ((p_m * 4.0) * p_m)));
	double t_1 = (4.0 * p_m) * p_m;
	double t_2 = x / sqrt(fma(x, x, t_1));
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((t_1 + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((((pow(t_0, 3.0) - -1.0) * 0.5) / fma(t_0, (fma(t_2, t_2, -1.0) / ((x / sqrt(fma((p_m * p_m), 4.0, (x * x)))) - -1.0)), 1.0)));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(fma(x, x, Float64(Float64(p_m * 4.0) * p_m))))
	t_1 = Float64(Float64(4.0 * p_m) * p_m)
	t_2 = Float64(x / sqrt(fma(x, x, t_1)))
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(t_1 + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(Float64(Float64((t_0 ^ 3.0) - -1.0) * 0.5) / fma(t_0, Float64(fma(t_2, t_2, -1.0) / Float64(Float64(x / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))) - -1.0)), 1.0)));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(x * x + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Sqrt[N[(x * x + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision] / N[(t$95$0 * N[(N[(t$95$2 * t$95$2 + -1.0), $MachinePrecision] / N[(N[(x / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{2} - 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}}, 1\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\color{blue}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{2} - 1}}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{2} - \color{blue}{1 \cdot 1}}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\color{blue}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, \color{blue}{x \cdot x}\right)}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      5. lift-fma.f64 N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot p\right) \cdot 4 + x \cdot x}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(p \cdot p\right) \cdot 4}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\left(p \cdot p\right)} \cdot 4}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{p \cdot \left(p \cdot 4\right)}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{x \cdot x + p \cdot \color{blue}{\left(p \cdot 4\right)}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\left(p \cdot 4\right) \cdot p}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\left(p \cdot 4\right) \cdot p}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{{\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\color{blue}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      14. pow2 N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\color{blue}{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} + \color{blue}{-1} \cdot 1}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} + \color{blue}{-1}}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, -1\right)}}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} - -1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, -1\right)}{\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}} - -1}, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p\_m \cdot 4\right) \cdot p\_m\right)}}\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(t\_0, t\_0 - 1, 1\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (fma x x (* (* p_m 4.0) p_m))))))
   (if (<=
        (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
        0.0)
     (/ (- p_m) x)
     (sqrt
      (/
       (fma (pow (/ x (sqrt (fma (* p_m p_m) 4.0 (* x x)))) 3.0) 0.5 0.5)
       (fma t_0 (- t_0 1.0) 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(fma(x, x, ((p_m * 4.0) * p_m)));
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((fma(pow((x / sqrt(fma((p_m * p_m), 4.0, (x * x)))), 3.0), 0.5, 0.5) / fma(t_0, (t_0 - 1.0), 1.0)));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(fma(x, x, Float64(Float64(p_m * 4.0) * p_m))))
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(fma((Float64(x / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))) ^ 3.0), 0.5, 0.5) / fma(t_0, Float64(t_0 - 1.0), 1.0)));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(x * x + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(N[Power[N[(x / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot \frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{2} \cdot \color{blue}{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\frac{\color{blue}{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      6. lower-fma.f64 99.7

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3}, 0.5, 0.5\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(p \cdot 4\right) \cdot p}}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(p \cdot 4\right) \cdot p}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot 4\right) \cdot p + x \cdot x}}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot 4\right) \cdot p} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{p \cdot \left(p \cdot 4\right)} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{p \cdot \color{blue}{\left(p \cdot 4\right)} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot p\right) \cdot 4} + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\left(p \cdot p\right)} \cdot 4 + x \cdot x}}\right)}^{3}, \frac{1}{2}, \frac{1}{2}\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

      15. lower-fma.f64 99.7

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}\right)}^{3}, 0.5, 0.5\right)}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{x}, \frac{p\_m \cdot p\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.05)
     (/ (- p_m) x)
     (if (<= t_0 0.8)
       (sqrt (fma (/ x p_m) 0.25 0.5))
       (fma (/ -0.5 x) (/ (* p_m p_m) x) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.05) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma((-0.5 / x), ((p_m * p_m) / x), 1.0);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = fma(Float64(-0.5 / x), Float64(Float64(p_m * p_m) / x), 1.0);
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(-0.5 / x), $MachinePrecision] * N[(N[(p$95$m * p$95$m), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.050000000000000003

   1. Initial program 22.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 22.9

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 22.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if 0.050000000000000003 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.80000000000000004

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{x}{p} + \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{p} \cdot \frac{1}{4}} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, \frac{1}{4}, \frac{1}{2}\right)}} \]

      4. lower-/.f64 98.9

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{p}}, 0.25, 0.5\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 100.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 100.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 100.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in p around 0

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{p}^{2}}{{x}^{2}} + 1} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot {p}^{2}}{{x}^{2}}} + 1 \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot {p}^{2}}{\color{blue}{x \cdot x}} + 1 \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x} \cdot \frac{{p}^{2}}{x}} + 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{x}, \frac{{p}^{2}}{x}, 1\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2}}{x}}, \frac{{p}^{2}}{x}, 1\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{x}, \color{blue}{\frac{{p}^{2}}{x}}, 1\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{x}, \frac{\color{blue}{p \cdot p}}{x}, 1\right) \]

   7. Applied rewrites 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.05:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{x}, \frac{p \cdot p}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.05)
     (/ (- p_m) x)
     (if (<= t_0 0.8) (sqrt (fma (/ x p_m) 0.25 0.5)) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.05) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.050000000000000003

   1. Initial program 22.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 22.9

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 22.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if 0.050000000000000003 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.80000000000000004

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{x}{p} + \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{p} \cdot \frac{1}{4}} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, \frac{1}{4}, \frac{1}{2}\right)}} \]

      4. lower-/.f64 98.9

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{p}}, 0.25, 0.5\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}} \]

   7. Taylor expanded in p around 0

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.05:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.05:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.05) (/ (- p_m) x) (if (<= t_0 0.8) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.05) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m =     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(p_m, x)
use fmin_fmax_functions
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p_m) * p_m) + (x * x)))))))
    if (t_0 <= 0.05d0) then
        tmp = -p_m / x
    else if (t_0 <= 0.8d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.05) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))))
	tmp = 0
	if t_0 <= 0.05:
		tmp = -p_m / x
	elif t_0 <= 0.8:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.05)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	tmp = 0.0;
	if (t_0 <= 0.05)
		tmp = -p_m / x;
	elseif (t_0 <= 0.8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.05], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.050000000000000003

   1. Initial program 22.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 22.9

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 22.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if 0.050000000000000003 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.80000000000000004

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 98.1%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}} \]

   7. Taylor expanded in p around 0

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.05:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.0)
   (/ (- p_m) x)
   (sqrt (fma (/ x (sqrt (fma (* p_m p_m) 4.0 (* x x)))) 0.5 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma((p_m * p_m), 4.0, (x * x)))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p\_m \cdot 4\right) \cdot p\_m\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.0)
   (/ (- p_m) x)
   (sqrt (fma (/ x (sqrt (fma x x (* (* p_m 4.0) p_m)))) 0.5 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma(x, x, ((p_m * 4.0) * p_m)))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(x, x, Float64(Float64(p_m * 4.0) * p_m)))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(x * x + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 99.7

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 99.7

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 99.7

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.0)
   (/ (- p_m) x)
   (sqrt (fma x (/ 0.5 (sqrt (fma (* p_m p_m) 4.0 (* x x)))) 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma(x, (0.5 / sqrt(fma((p_m * p_m), 4.0, (x * x)))), 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(x, Float64(0.5 / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))), 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.0

   1. Initial program 18.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 18.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 18.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \frac{0.5}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0.05:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{fma}\left(\frac{2}{x}, p\_m \cdot p\_m, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.05)
   (/ (- p_m) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (fma (/ 2.0 x) (* p_m p_m) x)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.05) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / fma((2.0 / x), (p_m * p_m), x)))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.05)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / fma(Float64(2.0 / x), Float64(p_m * p_m), x)))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.05], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(N[(2.0 / x), $MachinePrecision] * N[(p$95$m * p$95$m), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.050000000000000003

   1. Initial program 22.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 22.9

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x} + \left(4 \cdot p\right) \cdot p}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      10. lower-fma.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5, 0.5\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      13. lower-*.f64 22.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   4. Applied rewrites 22.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{p}{x}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 56.3

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

   7. Applied rewrites 56.3%

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

   if 0.050000000000000003 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.05:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{fma}\left(\frac{2}{x}, p \cdot p, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0.85:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.85)
   (sqrt 0.5)
   1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.85) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m =     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(p_m, x)
use fmin_fmax_functions
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p_m) * p_m) + (x * x))))))) <= 0.85d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.85) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.85:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.85], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.849999999999999978

   1. Initial program 75.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 69.1%

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

   if 0.849999999999999978 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

      4. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}} \]

   7. Taylor expanded in p around 0

      \[\leadsto \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.85:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.7% accurate, 58.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 1 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 1.0;
}

Fortran

p_m =     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(p_m, x)
use fmin_fmax_functions
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 1.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 1.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 1.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 82.8%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \frac{1}{2}}} \]

   3. lift-+.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \frac{1}{2}} \]

   4. flip3-+ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \frac{1}{2}} \]

   5. associate-*l/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left({1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}\right) \cdot \frac{1}{2}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

4. Applied rewrites 82.8%

   \[\leadsto \sqrt{\color{blue}{\frac{\left({\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}\right)}^{3} + 1\right) \cdot 0.5}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}} - 1, 1\right)}}} \]

6. Applied rewrites 82.8%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}} \]

7. Taylor expanded in p around 0

   \[\leadsto \color{blue}{1} \]
8. Step-by-step derivation

9. Applied rewrites 39.7%

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

10. Final simplification 39.7%

    \[\leadsto 1 \]
11. Add Preprocessing

Developer Target 1: 79.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2025015 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))