Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 10.3s

Alternatives: 8

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: 79.4% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 10^{-7}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left({t\_0}^{-0.25}, \frac{x}{{t\_0}^{0.25}}, 1\right) \cdot 0.5\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (fma (* p_m p_m) 4.0 (* x x))))
   (if (<=
        (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
        1e-7)
     (/ (- p_m) x)
     (exp
      (* (log (* (fma (pow t_0 -0.25) (/ x (pow t_0 0.25)) 1.0) 0.5)) 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = fma((p_m * p_m), 4.0, (x * x));
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 1e-7) {
		tmp = -p_m / x;
	} else {
		tmp = exp((log((fma(pow(t_0, -0.25), (x / pow(t_0, 0.25)), 1.0) * 0.5)) * 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = fma(Float64(p_m * p_m), 4.0, Float64(x * 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))))))) <= 1e-7)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = exp(Float64(log(Float64(fma((t_0 ^ -0.25), Float64(x / (t_0 ^ 0.25)), 1.0) * 0.5)) * 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $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], 1e-7], N[((-p$95$m) / x), $MachinePrecision], N[Exp[N[(N[Log[N[(N[(N[Power[t$95$0, -0.25], $MachinePrecision] * N[(x / N[Power[t$95$0, 0.25], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $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))))))) < 9.9999999999999995e-8

   1. Initial program 17.3%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 3.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 52.4

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

   7. Applied rewrites 52.4%

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

   if 9.9999999999999995e-8 < (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 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow1/2 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, \frac{1}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, 1\right)}} \cdot \sqrt{0.5} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.5

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, \frac{x}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, 1\right)}} \cdot \sqrt{0.5} \]

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 99.5

         \[\leadsto \sqrt{\mathsf{fma}\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{-0.25}}, \frac{x}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, 1\right)} \cdot \sqrt{0.5} \]

   8. Applied rewrites 99.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{-0.25}, \frac{x}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{0.25}}, 1\right)}} \cdot \sqrt{0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. pow1/2 N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 99.9%

       \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left({\left(\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)\right)}^{-0.25}, \frac{x}{{\left(\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)\right)}^{0.25}}, 1\right) \cdot 0.5\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.25}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \end{array} \end{array} \]

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 0.2)
     (/ (- p_m) x)
     (if (<= t_0 0.9)
       (sqrt (fma x (/ 0.25 p_m) 0.5))
       (- 1.0 (/ (* 0.5 (* p_m p_m)) (* x x)))))))

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 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.9) {
		tmp = sqrt(fma(x, (0.25 / p_m), 0.5));
	} else {
		tmp = 1.0 - ((0.5 * (p_m * p_m)) / (x * x));
	}
	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 <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.9)
		tmp = sqrt(fma(x, Float64(0.25 / p_m), 0.5));
	else
		tmp = Float64(1.0 - Float64(Float64(0.5 * Float64(p_m * p_m)) / Float64(x * x)));
	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, 0.2], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.9], N[Sqrt[N[(x * N[(0.25 / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[(N[(0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $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))))))) < 0.20000000000000001

   1. Initial program 19.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 5.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if 0.20000000000000001 < (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))))))) < 0.900000000000000022

   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 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   if 0.900000000000000022 < (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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 2\right)}}}, \frac{1}{2}\right)} \]

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{2}}, \frac{1}{2}\right)} \]

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. fabs-sqr N/A

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

      12. sqr-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. pow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}, 0.5\right)} \]

   7. Taylor expanded in p around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 98.1

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

   9. Applied rewrites 98.1%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.25}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.2)
     (/ (- p_m) x)
     (if (<= t_0 0.9) (sqrt (fma x (/ 0.25 p_m) 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 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.9) {
		tmp = sqrt(fma(x, (0.25 / p_m), 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 <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.9)
		tmp = sqrt(fma(x, Float64(0.25 / p_m), 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, 0.2], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.9], N[Sqrt[N[(x * N[(0.25 / p$95$m), $MachinePrecision] + 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))))))) < 0.20000000000000001

   1. Initial program 19.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 5.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if 0.20000000000000001 < (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))))))) < 0.900000000000000022

   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 98.6

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

   5. Applied rewrites 98.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   if 0.900000000000000022 < (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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 2\right)}}}, \frac{1}{2}\right)} \]

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{2}}, \frac{1}{2}\right)} \]

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. fabs-sqr N/A

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

      12. sqr-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. pow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}, 0.5\right)} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 97.1%

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

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.2) (/ (- p_m) x) (if (<= t_0 0.9) (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 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.9) {
		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 <= 0.2d0) then
        tmp = -p_m / x
    else if (t_0 <= 0.9d0) 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 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.9) {
		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 <= 0.2:
		tmp = -p_m / x
	elif t_0 <= 0.9:
		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 <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.9)
		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 <= 0.2)
		tmp = -p_m / x;
	elseif (t_0 <= 0.9)
		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, 0.2], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.9], 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))))))) < 0.20000000000000001

   1. Initial program 19.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 5.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if 0.20000000000000001 < (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))))))) < 0.900000000000000022

   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 97.9%

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

   if 0.900000000000000022 < (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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 2\right)}}}, \frac{1}{2}\right)} \]

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{2}}, \frac{1}{2}\right)} \]

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. fabs-sqr N/A

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

      12. sqr-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. pow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}, 0.5\right)} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 97.1%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 10^{-7}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\sqrt{\mathsf{fma}\left(p\_m \cdot 4, p\_m, x \cdot x\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

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)))))))
      1e-7)
   (/ (- p_m) x)
   (sqrt (fma x (/ 0.5 (sqrt (fma (* p_m 4.0) p_m (* 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))))))) <= 1e-7) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma(x, (0.5 / sqrt(fma((p_m * 4.0), p_m, (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))))))) <= 1e-7)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(x, Float64(0.5 / sqrt(fma(Float64(p_m * 4.0), p_m, 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], 1e-7], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m + 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))))))) < 9.9999999999999995e-8

   1. Initial program 17.3%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 3.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 52.4

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

   7. Applied rewrites 52.4%

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

   if 9.9999999999999995e-8 < (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 99.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

Alternative 6: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(p\_m \cdot p\_m, \frac{2}{x}, x\right)}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

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)))))))
      0.2)
   (/ (- p_m) x)
   (sqrt (fma (/ x (fma (* p_m p_m) (/ 2.0 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))))))) <= 0.2) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / fma((p_m * p_m), (2.0 / 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))))))) <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / fma(Float64(p_m * p_m), Float64(2.0 / 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], 0.2], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[(N[(p$95$m * p$95$m), $MachinePrecision] * N[(2.0 / x), $MachinePrecision] + x), $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))))))) < 0.20000000000000001

   1. Initial program 19.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 5.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]

   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. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 51.9

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

   7. Applied rewrites 51.9%

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

   if 0.20000000000000001 < (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 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 98.0

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

   7. Applied rewrites 98.0%

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

Alternative 7: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0.86:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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)))))))
      0.86)
   (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))))))) <= 0.86) {
		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))))))) <= 0.86d0) 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))))))) <= 0.86) {
		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))))))) <= 0.86:
		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))))))) <= 0.86)
		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))))))) <= 0.86)
		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], 0.86], 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))))))) < 0.859999999999999987

   1. Initial program 72.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 66.6%

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

   if 0.859999999999999987 < (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-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqr-pow N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 2\right)}}}, \frac{1}{2}\right)} \]

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{2}}, \frac{1}{2}\right)} \]

      9. metadata-eval N/A

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

      10. pow2 N/A

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

      11. fabs-sqr N/A

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

      12. sqr-pow N/A

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

      13. lift-pow.f64 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. pow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. pow1/2 N/A

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

      20. lift-fma.f64 N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}, 0.5\right)} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 97.1%

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

Alternative 8: 35.1% 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 79.2%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
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-lft-in N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqr-pow N/A

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

   11. times-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

4. Applied rewrites 75.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, 0.5\right)}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 2\right)}}}, \frac{1}{2}\right)} \]

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{\frac{1}{4}}}, \frac{\frac{1}{2}}{{\left({\left(\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{2}}, \frac{1}{2}\right)} \]

   9. metadata-eval N/A

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

   10. pow2 N/A

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

   11. fabs-sqr N/A

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

   12. sqr-pow N/A

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

   13. lift-pow.f64 N/A

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

   14. rem-sqrt-square-rev N/A

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

   15. pow2 N/A

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

   16. lift-pow.f64 N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. pow1/2 N/A

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

   20. lift-fma.f64 N/A

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

6. Applied rewrites 75.6%

   \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{{\left(\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)\right)}^{0.25}}, \frac{0.5}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}, 0.5\right)} \]

7. Taylor expanded in p around 0

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

9. Applied rewrites 34.6%

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

Developer Target 1: 79.4% 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 2024356 
(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))))))))