Given's Rotation SVD example

Percentage Accurate: 79.2% → 99.7%

Time: 5.3s

Alternatives: 6

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 14.8

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

   4. Applied rewrites 14.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. flip-+ N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}\right)}^{2} - 1}{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} - 1}}} \]

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 52.3

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

   9. Applied rewrites 52.3%

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

   10. Taylor expanded in p around 0

       \[\leadsto -\frac{p}{x} \]
   11. Step-by-step derivation

       1. lower-/.f64 52.5

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

   12. Applied rewrites 52.5%

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

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

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 87.1%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.4:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p\_m \cdot p\_m}{x \cdot x} \cdot 0.5, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.4)
     (- (/ p_m x))
     (if (<= t_0 0.9)
       (sqrt (fma (/ x p_m) 0.25 0.5))
       (fma (* (/ (* p_m p_m) (* x x)) 0.5) -1.0 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 <= 0.4) {
		tmp = -(p_m / x);
	} else if (t_0 <= 0.9) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma((((p_m * p_m) / (x * x)) * 0.5), -1.0, 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 <= 0.4)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 0.9)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = fma(Float64(Float64(Float64(p_m * p_m) / Float64(x * x)) * 0.5), -1.0, 1.0);
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[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, 0.4], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 0.9], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.40000000000000002

   1. Initial program 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 16.9

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

   4. Applied rewrites 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. flip-+ N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 16.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}\right)}^{2} - 1}{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} - 1}}} \]

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 51.7

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in p around 0

       \[\leadsto -\frac{p}{x} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.9

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

   12. Applied rewrites 51.9%

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

   if 0.40000000000000002 < (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

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

      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 98.3

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

   5. Applied rewrites 98.3%

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

   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. 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%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.4:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p \cdot p}{x \cdot x} \cdot 0.5, -1, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.4:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.4)
     (- (/ p_m x))
     (if (<= t_0 0.9) (sqrt (fma (/ x p_m) 0.25 0.5)) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.4) {
		tmp = -(p_m / x);
	} else if (t_0 <= 0.9) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.4)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 0.9)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.4], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 0.9], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.40000000000000002

   1. Initial program 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 16.9

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

   4. Applied rewrites 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. flip-+ N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 16.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}\right)}^{2} - 1}{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} - 1}}} \]

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 51.7

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in p around 0

       \[\leadsto -\frac{p}{x} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.9

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

   12. Applied rewrites 51.9%

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

   if 0.40000000000000002 < (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

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

      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 98.3

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

   5. Applied rewrites 98.3%

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

   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. 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 99.9

         \[\leadsto 1 \]

   5. Applied rewrites 99.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.4:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.9:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.4:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.9:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 0.4) (- (/ 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 <= 0.4) {
		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 <= 0.4d0) 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 <= 0.4) {
		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 <= 0.4:
		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 <= 0.4)
		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 <= 0.4)
		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, 0.4], (-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))))))) < 0.40000000000000002

   1. Initial program 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 16.9

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

   4. Applied rewrites 16.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. flip-+ N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 16.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{\left(\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}\right)}^{2} - 1}{\frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} - 1}}} \]

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 51.7

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in p around 0

       \[\leadsto -\frac{p}{x} \]
   11. Step-by-step derivation

       1. lower-/.f64 51.9

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

   12. Applied rewrites 51.9%

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

   if 0.40000000000000002 < (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 97.8%

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

   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. 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 99.9

         \[\leadsto 1 \]

   5. Applied rewrites 99.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.4:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.9:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.8× speedup

Math

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

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 70.5%

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

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 65.1%

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

   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. 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 99.9

         \[\leadsto 1 \]

   5. Applied rewrites 99.9%

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

Alternative 6: 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 76.9%

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

3. 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 33.4

      \[\leadsto 1 \]

5. Applied rewrites 33.4%

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

Developer Target 1: 79.2% 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 2025073 
(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))))))))