Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.8%

Time: 4.4s

Alternatives: 7

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: 79.6% 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, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{p\_m \cdot p\_m}{x \cdot x}, 1\right) \cdot p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot p\_m, 4, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      5e-8)
   (/ (* (fma -1.5 (/ (* p_m p_m) (* x x)) 1.0) p_m) (- x))
   (sqrt (fma (/ x (sqrt (fma (* p_m p_m) 4.0 (* x x)))) 0.5 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 5e-8) {
		tmp = (fma(-1.5, ((p_m * p_m) / (x * x)), 1.0) * p_m) / -x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma((p_m * p_m), 4.0, (x * x)))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 5e-8)
		tmp = Float64(Float64(fma(-1.5, Float64(Float64(p_m * p_m) / Float64(x * x)), 1.0) * p_m) / Float64(-x));
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(Float64(p_m * p_m), 4.0, Float64(x * x)))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-8], N[(N[(N[(-1.5 * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * p$95$m), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 4.9999999999999998e-8

   1. Initial program 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      4. +-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      6. pow2 N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}\right) \]

   7. Applied rewrites 57.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{\frac{{p}^{4} \cdot -12}{p}}{x \cdot x}, 0.125, p\right)}{x}} \]

   8. Taylor expanded in p around 0

      \[\leadsto -\frac{p \cdot \left(1 + \frac{-3}{2} \cdot \frac{{p}^{2}}{{x}^{2}}\right)}{x} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

      9. lift-*.f64 57.6

         \[\leadsto -\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

   10. Applied rewrites 57.6%

       \[\leadsto -\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

   if 4.9999999999999998e-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 99.8%

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

      1. lift-*.f64 N/A

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

      2. 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. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot p, 4, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ t_1 := \frac{p\_m \cdot p\_m}{x \cdot x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, t\_1, 1\right) \cdot p\_m}{-x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x))))))))
        (t_1 (/ (* p_m p_m) (* x x))))
   (if (<= t_0 5e-8)
     (/ (* (fma -1.5 t_1 1.0) p_m) (- x))
     (if (<= t_0 0.8) (sqrt (fma (/ x p_m) 0.25 0.5)) (fma t_1 -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 t_1 = (p_m * p_m) / (x * x);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = (fma(-1.5, t_1, 1.0) * p_m) / -x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma(t_1, -0.5, 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)))))))
	t_1 = Float64(Float64(p_m * p_m) / Float64(x * x))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(Float64(fma(-1.5, t_1, 1.0) * p_m) / Float64(-x));
	elseif (t_0 <= 0.8)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = fma(t_1, -0.5, 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]}, Block[{t$95$1 = N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[(N[(N[(-1.5 * t$95$1 + 1.0), $MachinePrecision] * p$95$m), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * -0.5 + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 4.9999999999999998e-8

   1. Initial program 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      4. +-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      6. pow2 N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}\right) \]

   7. Applied rewrites 57.6%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{\frac{{p}^{4} \cdot -12}{p}}{x \cdot x}, 0.125, p\right)}{x}} \]

   8. Taylor expanded in p around 0

      \[\leadsto -\frac{p \cdot \left(1 + \frac{-3}{2} \cdot \frac{{p}^{2}}{{x}^{2}}\right)}{x} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

      9. lift-*.f64 57.6

         \[\leadsto -\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

   10. Applied rewrites 57.6%

       \[\leadsto -\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{x} \]

   if 4.9999999999999998e-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))))))) < 0.80000000000000004

   1. Initial program 99.7%

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

   3. 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. times-frac N/A

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

      7. sqrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot p}{x \cdot x}, 1\right) \cdot p}{-x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p \cdot p}{x \cdot x}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p\_m \cdot p\_m}{x \cdot x}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 5e-8)
     (/ (- p_m) x)
     (if (<= t_0 0.8)
       (sqrt (fma (/ x p_m) 0.25 0.5))
       (fma (/ (* p_m p_m) (* x x)) -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 <= 5e-8) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma(((p_m * p_m) / (x * x)), -0.5, 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 <= 5e-8)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = fma(Float64(Float64(p_m * p_m) / Float64(x * x)), -0.5, 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, 5e-8], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 4.9999999999999998e-8

   1. Initial program 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x around -inf

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      4. +-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      6. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if 4.9999999999999998e-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))))))) < 0.80000000000000004

   1. Initial program 99.7%

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

   3. 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. times-frac N/A

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

      7. sqrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p \cdot p}{x \cdot x}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p\_m \cdot p\_m}{x \cdot x}, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 5e-8)
     (/ (- p_m) x)
     (if (<= t_0 0.8) (sqrt 0.5) (fma (/ (* p_m p_m) (* x x)) -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 <= 5e-8) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(0.5);
	} else {
		tmp = fma(((p_m * p_m) / (x * x)), -0.5, 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 <= 5e-8)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(0.5);
	else
		tmp = fma(Float64(Float64(p_m * p_m) / Float64(x * x)), -0.5, 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, 5e-8], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[0.5], $MachinePrecision], N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 4.9999999999999998e-8

   1. Initial program 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x around -inf

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      4. +-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      6. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if 4.9999999999999998e-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))))))) < 0.80000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. times-frac N/A

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

      7. sqrt-undiv N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p \cdot p}{x \cdot x}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 5e-8) (/ (- p_m) x) (if (<= t_0 0.8) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(p_m, x)
use fmin_fmax_functions
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p_m) * p_m) + (x * x)))))))
    if (t_0 <= 5d-8) then
        tmp = -p_m / x
    else if (t_0 <= 0.8d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.8) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))))
	tmp = 0
	if t_0 <= 5e-8:
		tmp = -p_m / x
	elif t_0 <= 0.8:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	tmp = 0.0;
	if (t_0 <= 5e-8)
		tmp = -p_m / x;
	elseif (t_0 <= 0.8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 4.9999999999999998e-8

   1. Initial program 17.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x around -inf

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      4. +-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      6. pow2 N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      7. distribute-lft-in N/A

         \[\leadsto -1 \cdot \frac{p}{x} \]

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if 4.9999999999999998e-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))))))) < 0.80000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 98.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      3. metadata-eval 99.1

         \[\leadsto 1 \]

   5. Applied rewrites 99.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

p_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.7%

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

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 74.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      3. metadata-eval 99.1

         \[\leadsto 1 \]

   5. Applied rewrites 99.1%

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

Alternative 7: 36.2% 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 84.8%

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

3. Taylor expanded in p around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   3. metadata-eval 39.8

      \[\leadsto 1 \]

5. Applied rewrites 39.8%

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

Developer Target 1: 79.6% 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 2025051 
(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))))))))