Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.9%

Time: 5.5s

Alternatives: 11

Speedup: 0.5×

Specification

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

Math

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

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

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

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\left|p\right|}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{0.5}{x \cdot x} + \frac{0.5}{\sqrt{\mathsf{fma}\left(4, p \cdot p, x \cdot x\right)} \cdot x}}{\frac{1}{x \cdot x}}}\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))) 2e-8)
   (/ (- (fabs p)) x)
   (sqrt
    (/
     (+ (/ 0.5 (* x x)) (/ 0.5 (* (sqrt (fma 4.0 (* p p) (* x x))) x)))
     (/ 1.0 (* x x))))))

C

double code(double p, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x))))))) <= 2e-8) {
		tmp = -fabs(p) / x;
	} else {
		tmp = sqrt((((0.5 / (x * x)) + (0.5 / (sqrt(fma(4.0, (p * p), (x * x))) * x))) / (1.0 / (x * x))));
	}
	return tmp;
}

Julia

function code(p, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x))))))) <= 2e-8)
		tmp = Float64(Float64(-abs(p)) / x);
	else
		tmp = sqrt(Float64(Float64(Float64(0.5 / Float64(x * x)) + Float64(0.5 / Float64(sqrt(fma(4.0, Float64(p * p), Float64(x * x))) * x))) / Float64(1.0 / Float64(x * x))));
	end
	return tmp
end

Wolfram

code[p_, x_] := If[LessEqual[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], 2e-8], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[Sqrt[N[(4.0 * N[(p * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x * x), $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))))))) < 2e-8

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 2e-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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. common-denominator N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 79.6%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[p_, x_] := If[LessEqual[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], 5e-5], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(N[Sqrt[N[(N[(4.0 * p), $MachinePrecision] * p + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 5.00000000000000024e-5

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 5.00000000000000024e-5 < (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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[p_, x_] := If[LessEqual[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], 5e-5], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(N[Sqrt[N[(1.0 / N[(N[(p * p), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $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))))))) < 5.00000000000000024e-5

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 5.00000000000000024e-5 < (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 79.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/r* N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. *-inverses N/A

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

      9. mult-flip N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 51.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 79.1%

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

Alternative 4: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[p_, x_] := If[LessEqual[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], 5e-5], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(N[(p * 4.0), $MachinePrecision] * p + 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))))))) < 5.00000000000000024e-5

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 5.00000000000000024e-5 < (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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[p_, x_] := If[LessEqual[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], 5e-5], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[(p * 4.0), $MachinePrecision] * p + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x + 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))))))) < 5.00000000000000024e-5

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 5.00000000000000024e-5 < (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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

   3. Applied rewrites 76.5%

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

Alternative 6: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot \left|p\right|\right) \cdot \left|p\right| + x \cdot x}}\right)}\\ t_1 := 2 \cdot \left|p\right|\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{-\left|\left|p\right|\right|}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{\frac{\left(t\_1 + x\right) \cdot 0.5}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           0.5
           (+ 1.0 (/ x (sqrt (+ (* (* 4.0 (fabs p)) (fabs p)) (* x x))))))))
        (t_1 (* 2.0 (fabs p))))
   (if (<= t_0 0.002)
     (/ (- (fabs (fabs p))) x)
     (if (<= t_0 0.8) (sqrt (/ (* (+ t_1 x) 0.5) t_1)) 1.0))))

C

double code(double p, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * fabs(p)) * fabs(p)) + (x * x)))))));
	double t_1 = 2.0 * fabs(p);
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -fabs(fabs(p)) / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt((((t_1 + x) * 0.5) / t_1));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * abs(p)) * abs(p)) + (x * x)))))))
    t_1 = 2.0d0 * abs(p)
    if (t_0 <= 0.002d0) then
        tmp = -abs(abs(p)) / x
    else if (t_0 <= 0.8d0) then
        tmp = sqrt((((t_1 + x) * 0.5d0) / t_1))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * Math.abs(p)) * Math.abs(p)) + (x * x)))))));
	double t_1 = 2.0 * Math.abs(p);
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -Math.abs(Math.abs(p)) / x;
	} else if (t_0 <= 0.8) {
		tmp = Math.sqrt((((t_1 + x) * 0.5) / t_1));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * math.fabs(p)) * math.fabs(p)) + (x * x)))))))
	t_1 = 2.0 * math.fabs(p)
	tmp = 0
	if t_0 <= 0.002:
		tmp = -math.fabs(math.fabs(p)) / x
	elif t_0 <= 0.8:
		tmp = math.sqrt((((t_1 + x) * 0.5) / t_1))
	else:
		tmp = 1.0
	return tmp

Julia

function code(p, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * abs(p)) * abs(p)) + Float64(x * x)))))))
	t_1 = Float64(2.0 * abs(p))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(-abs(abs(p))) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(Float64(Float64(Float64(t_1 + x) * 0.5) / t_1));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * abs(p)) * abs(p)) + (x * x)))))));
	t_1 = 2.0 * abs(p);
	tmp = 0.0;
	if (t_0 <= 0.002)
		tmp = -abs(abs(p)) / x;
	elseif (t_0 <= 0.8)
		tmp = sqrt((((t_1 + x) * 0.5) / t_1));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * N[Abs[p], $MachinePrecision]), $MachinePrecision] * N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[((-N[Abs[N[Abs[p], $MachinePrecision]], $MachinePrecision]) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(N[(N[(t$95$1 + x), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$1), $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))))))) < 2e-3

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 2e-3 < (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 79.6%

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in p around inf

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

      1. lower-*.f64 17.7

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

   6. Applied rewrites 17.7%

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

   7. Taylor expanded in p around inf

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

      1. lower-*.f64 51.0

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

   9. Applied rewrites 51.0%

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

   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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.4%

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

Alternative 7: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot \left|p\right|\right) \cdot \left|p\right| + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{-\left|\left|p\right|\right|}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{\left|p\right|}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           0.5
           (+ 1.0 (/ x (sqrt (+ (* (* 4.0 (fabs p)) (fabs p)) (* x x)))))))))
   (if (<= t_0 0.002)
     (/ (- (fabs (fabs p))) x)
     (if (<= t_0 0.8) (sqrt (+ 0.5 (* 0.25 (/ x (fabs p))))) 1.0))))

C

double code(double p, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * fabs(p)) * fabs(p)) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -fabs(fabs(p)) / x;
	} else if (t_0 <= 0.8) {
		tmp = sqrt((0.5 + (0.25 * (x / fabs(p)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * abs(p)) * abs(p)) + (x * x)))))))
    if (t_0 <= 0.002d0) then
        tmp = -abs(abs(p)) / x
    else if (t_0 <= 0.8d0) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / abs(p)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * Math.abs(p)) * Math.abs(p)) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -Math.abs(Math.abs(p)) / x;
	} else if (t_0 <= 0.8) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / Math.abs(p)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * math.fabs(p)) * math.fabs(p)) + (x * x)))))))
	tmp = 0
	if t_0 <= 0.002:
		tmp = -math.fabs(math.fabs(p)) / x
	elif t_0 <= 0.8:
		tmp = math.sqrt((0.5 + (0.25 * (x / math.fabs(p)))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(p, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * abs(p)) * abs(p)) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(-abs(abs(p))) / x);
	elseif (t_0 <= 0.8)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / abs(p)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * abs(p)) * abs(p)) + (x * x)))))));
	tmp = 0.0;
	if (t_0 <= 0.002)
		tmp = -abs(abs(p)) / x;
	elseif (t_0 <= 0.8)
		tmp = sqrt((0.5 + (0.25 * (x / abs(p)))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * N[Abs[p], $MachinePrecision]), $MachinePrecision] * N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[((-N[Abs[N[Abs[p], $MachinePrecision]], $MachinePrecision]) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))))) < 2e-3

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 2e-3 < (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 79.6%

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

   2. Taylor expanded in p around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 51.0

         \[\leadsto \sqrt{0.5 + 0.25 \cdot \frac{x}{\color{blue}{p}}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]

   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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.4%

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

Alternative 8: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot \left|p\right|\right) \cdot \left|p\right| + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{-\left|\left|p\right|\right|}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\left|p\right|}, 0.17677669529663687, 0.7071067811865476\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           0.5
           (+ 1.0 (/ x (sqrt (+ (* (* 4.0 (fabs p)) (fabs p)) (* x x)))))))))
   (if (<= t_0 0.002)
     (/ (- (fabs (fabs p))) x)
     (if (<= t_0 0.8)
       (fma (/ x (fabs p)) 0.17677669529663687 0.7071067811865476)
       1.0))))

C

double code(double p, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * fabs(p)) * fabs(p)) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -fabs(fabs(p)) / x;
	} else if (t_0 <= 0.8) {
		tmp = fma((x / fabs(p)), 0.17677669529663687, 0.7071067811865476);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(p, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * abs(p)) * abs(p)) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(-abs(abs(p))) / x);
	elseif (t_0 <= 0.8)
		tmp = fma(Float64(x / abs(p)), 0.17677669529663687, 0.7071067811865476);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * N[Abs[p], $MachinePrecision]), $MachinePrecision] * N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[((-N[Abs[N[Abs[p], $MachinePrecision]], $MachinePrecision]) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(N[(x / N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.17677669529663687 + 0.7071067811865476), $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))))))) < 2e-3

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 2e-3 < (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 79.6%

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

   2. Taylor expanded in p around inf

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

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \color{blue}{\frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{\color{blue}{p \cdot \sqrt{\frac{1}{2}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{p \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      6. lower-sqrt.f64 51.0

         \[\leadsto \sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}}} \]

   5. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

   6. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6369051672525773}{9007199254740992} + \color{blue}{\frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} + \color{blue}{\frac{6369051672525773}{9007199254740992}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} + \frac{6369051672525773}{9007199254740992} \]

      4. *-commutative N/A

         \[\leadsto \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \cdot \frac{1}{8} + \frac{6369051672525773}{9007199254740992} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \cdot \frac{1}{8} + \frac{6369051672525773}{9007199254740992} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{x \cdot \frac{1}{8}}{p \cdot \frac{6369051672525773}{9007199254740992}} + \frac{6369051672525773}{9007199254740992} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \frac{1}{8}}{p \cdot \frac{6369051672525773}{9007199254740992}} + \frac{6369051672525773}{9007199254740992} \]

      8. times-frac N/A

         \[\leadsto \frac{x}{p} \cdot \frac{\frac{1}{8}}{\frac{6369051672525773}{9007199254740992}} + \frac{6369051672525773}{9007199254740992} \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{p}, \color{blue}{\frac{\frac{1}{8}}{\frac{6369051672525773}{9007199254740992}}}, \frac{6369051672525773}{9007199254740992}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{p}, \frac{\color{blue}{\frac{1}{8}}}{\frac{6369051672525773}{9007199254740992}}, \frac{6369051672525773}{9007199254740992}\right) \]

      11. metadata-eval 51.0

          \[\leadsto \mathsf{fma}\left(\frac{x}{p}, 0.17677669529663687, 0.7071067811865476\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{p}, \color{blue}{0.17677669529663687}, 0.7071067811865476\right) \]

   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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.4%

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

Alternative 9: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{-\left|p\right|}{x}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;0.7071067811865476\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x)))))))))
   (if (<= t_0 0.002)
     (/ (- (fabs p)) x)
     (if (<= t_0 0.8) 0.7071067811865476 1.0))))

C

double code(double p, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -fabs(p) / x;
	} else if (t_0 <= 0.8) {
		tmp = 0.7071067811865476;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
    if (t_0 <= 0.002d0) then
        tmp = -abs(p) / x
    else if (t_0 <= 0.8d0) then
        tmp = 0.7071067811865476d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.002) {
		tmp = -Math.abs(p) / x;
	} else if (t_0 <= 0.8) {
		tmp = 0.7071067811865476;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))
	tmp = 0
	if t_0 <= 0.002:
		tmp = -math.fabs(p) / x
	elif t_0 <= 0.8:
		tmp = 0.7071067811865476
	else:
		tmp = 1.0
	return tmp

Julia

function code(p, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(-abs(p)) / x);
	elseif (t_0 <= 0.8)
		tmp = 0.7071067811865476;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
	tmp = 0.0;
	if (t_0 <= 0.002)
		tmp = -abs(p) / x;
	elseif (t_0 <= 0.8)
		tmp = 0.7071067811865476;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.002], N[((-N[Abs[p], $MachinePrecision]) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.8], 0.7071067811865476, 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))))))) < 2e-3

   1. Initial program 79.6%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 17.5

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

   4. Applied rewrites 17.5%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 17.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. rem-sqrt-square N/A

          \[\leadsto \frac{-\left|p\right|}{x} \]

      11. lower-fabs.f64 26.4

          \[\leadsto \frac{-\left|p\right|}{x} \]

   6. Applied rewrites 26.4%

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

   if 2e-3 < (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 79.6%

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

   2. Taylor expanded in p around inf

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

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \color{blue}{\frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{\color{blue}{p \cdot \sqrt{\frac{1}{2}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{p \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      6. lower-sqrt.f64 51.0

         \[\leadsto \sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}}} \]

   5. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

   6. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \]

   7. Taylor expanded in p around inf

      \[\leadsto \frac{6369051672525773}{9007199254740992} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.7%

      \[\leadsto 0.7071067811865476 \]

   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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.4%

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

Alternative 10: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.99:\\ \;\;\;\;0.7071067811865476\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))) 0.99)
   0.7071067811865476
   1.0))

C

double code(double p, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x))))))) <= 0.99) {
		tmp = 0.7071067811865476;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x))))))) <= 0.99d0) then
        tmp = 0.7071067811865476d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x))))))) <= 0.99) {
		tmp = 0.7071067811865476;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x))))))) <= 0.99:
		tmp = 0.7071067811865476
	else:
		tmp = 1.0
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x))))))) <= 0.99)
		tmp = 0.7071067811865476;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x))))))) <= 0.99)
		tmp = 0.7071067811865476;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[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], 0.99], 0.7071067811865476, 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))))))) < 0.98999999999999999

   1. Initial program 79.6%

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

   2. Taylor expanded in p around inf

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

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \color{blue}{\frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{\color{blue}{p \cdot \sqrt{\frac{1}{2}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{p \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      6. lower-sqrt.f64 51.0

         \[\leadsto \sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}}} \]

   5. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

   6. Evaluated real constant 51.0%

      \[\leadsto \frac{6369051672525773}{9007199254740992} + \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \]

   7. Taylor expanded in p around inf

      \[\leadsto \frac{6369051672525773}{9007199254740992} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.7%

      \[\leadsto 0.7071067811865476 \]

   if 0.98999999999999999 < (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 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 79.6

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

   3. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 79.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-/r* N/A

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

      9. add-to-fraction N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 36.4%

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

Alternative 11: 55.7% accurate, 25.2× speedup

Math

\[0.7071067811865476 \]

FPCore

(FPCore (p x) :precision binary64 0.7071067811865476)

C

double code(double p, double x) {
	return 0.7071067811865476;
}

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 = 0.7071067811865476d0
end function

Java

public static double code(double p, double x) {
	return 0.7071067811865476;
}

Python

def code(p, x):
	return 0.7071067811865476

Julia

function code(p, x)
	return 0.7071067811865476
end

MATLAB

function tmp = code(p, x)
	tmp = 0.7071067811865476;
end

Wolfram

code[p_, x_] := 0.7071067811865476

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in p around inf

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

   1. lower-+.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2}} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \color{blue}{\frac{x}{p \cdot \sqrt{\frac{1}{2}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{\color{blue}{p \cdot \sqrt{\frac{1}{2}}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2}} + \frac{1}{8} \cdot \frac{x}{p \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

   6. lower-sqrt.f64 51.0

      \[\leadsto \sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}} \]

4. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\sqrt{0.5} + 0.125 \cdot \frac{x}{p \cdot \sqrt{0.5}}} \]

5. Evaluated real constant 51.0%

   \[\leadsto \frac{6369051672525773}{9007199254740992} + \color{blue}{\frac{1}{8}} \cdot \frac{x}{p \cdot \sqrt{\frac{1}{2}}} \]

6. Evaluated real constant 51.0%

   \[\leadsto \frac{6369051672525773}{9007199254740992} + \frac{1}{8} \cdot \frac{x}{p \cdot \frac{6369051672525773}{9007199254740992}} \]

7. Taylor expanded in p around inf

   \[\leadsto \frac{6369051672525773}{9007199254740992} \]
8. Step-by-step derivation

9. Applied rewrites 55.7%

   \[\leadsto 0.7071067811865476 \]
10. Add Preprocessing

Developer Target 1: 79.6% accurate, 0.8× speedup

Math

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

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 2025175 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform c (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))))))))