1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))

Percentage Accurate: 44.9% → 81.1%

Time: 7.8s

Alternatives: 11

Speedup: 12.5×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 44.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \end{array} \]

FPCore

(FPCore (p r q)
 :precision binary64
 (*
  (/ 1.0 2.0)
  (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))

C

double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((fabs(p) + fabs(r)) + sqrt((pow((p - r), 2.0) + (4.0 * pow(q, 2.0)))));
}

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, r, q)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q
    code = (1.0d0 / 2.0d0) * ((abs(p) + abs(r)) + sqrt((((p - r) ** 2.0d0) + (4.0d0 * (q ** 2.0d0)))))
end function

Java

public static double code(double p, double r, double q) {
	return (1.0 / 2.0) * ((Math.abs(p) + Math.abs(r)) + Math.sqrt((Math.pow((p - r), 2.0) + (4.0 * Math.pow(q, 2.0)))));
}

Python

def code(p, r, q):
	return (1.0 / 2.0) * ((math.fabs(p) + math.fabs(r)) + math.sqrt((math.pow((p - r), 2.0) + (4.0 * math.pow(q, 2.0)))))

Julia

function code(p, r, q)
	return Float64(Float64(1.0 / 2.0) * Float64(Float64(abs(p) + abs(r)) + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q ^ 2.0))))))
end

MATLAB

function tmp = code(p, r, q)
	tmp = (1.0 / 2.0) * ((abs(p) + abs(r)) + sqrt((((p - r) ^ 2.0) + (4.0 * (q ^ 2.0)))));
end

Wolfram

code[p_, r_, q_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 9.2 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 9.2e+48)
   (* 0.5 (+ (+ r (fabs p)) (- (fabs r) p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 9.2e+48) {
		tmp = 0.5 * ((r + fabs(p)) + (fabs(r) - p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 9.2e+48)
		tmp = Float64(0.5 * Float64(Float64(r + abs(p)) + Float64(abs(r) - p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 9.2e+48], N[(0.5 * N[(N[(r + N[Abs[p], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 9.2000000000000001e48

   1. Initial program 53.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   if 9.2000000000000001e48 < q

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 74.6

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 1.2 \cdot 10^{-63}:\\ \;\;\;\;\left(\left(\left|r\right| - p\right) + \left|p\right|\right) \cdot 0.5\\ \mathbf{elif}\;r \leq 1.2 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 1.2e-63)
   (* (+ (- (fabs r) p) (fabs p)) 0.5)
   (if (<= r 1.2e+73)
     (fma 0.5 (+ (fabs r) (fabs p)) q_m)
     (* 0.5 (+ (+ r (fabs r)) (fabs p))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 1.2e-63) {
		tmp = ((fabs(r) - p) + fabs(p)) * 0.5;
	} else if (r <= 1.2e+73) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = 0.5 * ((r + fabs(r)) + fabs(p));
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 1.2e-63)
		tmp = Float64(Float64(Float64(abs(r) - p) + abs(p)) * 0.5);
	elseif (r <= 1.2e+73)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(0.5 * Float64(Float64(r + abs(r)) + abs(p)));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 1.2e-63], N[(N[(N[(N[Abs[r], $MachinePrecision] - p), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[r, 1.2e+73], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(0.5 * N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.2e-63

   1. Initial program 50.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.3%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 24.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 10.8%

       \[\leadsto 0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right) \]

   12. Taylor expanded in r around 0

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

   14. Applied rewrites 28.6%

       \[\leadsto \left(\left(\left|r\right| - p\right) + \left|p\right|\right) \cdot \color{blue}{0.5} \]

   if 1.2e-63 < r < 1.20000000000000001e73

   1. Initial program 62.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 29.9

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

   if 1.20000000000000001e73 < r

   1. Initial program 36.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 89.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 86.1%

       \[\leadsto 0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 59.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 5e+48)
   (* 0.5 (+ (+ r (fabs r)) (fabs p)))
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 5e+48) {
		tmp = 0.5 * ((r + fabs(r)) + fabs(p));
	} else {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 5e+48)
		tmp = Float64(0.5 * Float64(Float64(r + abs(r)) + abs(p)));
	else
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 5e+48], N[(0.5 * N[(N[(r + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 4.99999999999999973e48

   1. Initial program 53.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 34.8%

       \[\leadsto 0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right) \]

   if 4.99999999999999973e48 < q

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 74.6

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 3 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(r - p\right) + r\right) + p\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 3e+74)
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)
   (* 0.5 (+ (+ (- r p) r) p))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 3e+74) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = 0.5 * (((r - p) + r) + p);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 3e+74)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(r - p) + r) + p));
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 3e+74], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(0.5 * N[(N[(N[(r - p), $MachinePrecision] + r), $MachinePrecision] + p), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3e74

   1. Initial program 52.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.2

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

   if 3e74 < r

   1. Initial program 36.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 89.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.7%

       \[\leadsto 0.5 \cdot \left(\left(\left(r - p\right) + r\right) + p\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 5.2 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left|r\right| + \left|p\right|, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + p\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= r 5.2e+74)
   (fma 0.5 (+ (fabs r) (fabs p)) q_m)
   (* (+ (+ (fabs r) r) p) 0.5)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 5.2e+74) {
		tmp = fma(0.5, (fabs(r) + fabs(p)), q_m);
	} else {
		tmp = ((fabs(r) + r) + p) * 0.5;
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 5.2e+74)
		tmp = fma(0.5, Float64(abs(r) + abs(p)), q_m);
	else
		tmp = Float64(Float64(Float64(abs(r) + r) + p) * 0.5);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 5.2e+74], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] + q$95$m), $MachinePrecision], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.2000000000000001e74

   1. Initial program 52.0%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 28.2

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

   if 5.2000000000000001e74 < r

   1. Initial program 36.4%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 89.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 86.1%

       \[\leadsto 0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 85.4%

       \[\leadsto \color{blue}{\left(\left(\left|r\right| + r\right) + p\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\left(\left(\left|r\right| + r\right) + p\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 5e+48) (* (+ (+ (fabs r) r) p) 0.5) (fma 0.5 r q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 5e+48) {
		tmp = ((fabs(r) + r) + p) * 0.5;
	} else {
		tmp = fma(0.5, r, q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 5e+48)
		tmp = Float64(Float64(Float64(abs(r) + r) + p) * 0.5);
	else
		tmp = fma(0.5, r, q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 5e+48], N[(N[(N[(N[Abs[r], $MachinePrecision] + r), $MachinePrecision] + p), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * r + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 4.99999999999999973e48

   1. Initial program 53.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in r around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(r + \left|p\right|\right) + \left(\left|r\right| - p\right)\right)} \]

   9. Taylor expanded in p around 0

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

   11. Applied rewrites 34.8%

       \[\leadsto 0.5 \cdot \left(\left(r + \left|r\right|\right) + \left|p\right|\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 31.1%

       \[\leadsto \color{blue}{\left(\left(\left|r\right| + r\right) + p\right) \cdot 0.5} \]

   if 4.99999999999999973e48 < q

   1. Initial program 29.9%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 74.6

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

   10. Applied rewrites 71.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r + p, 0.5, q\right)} \]

   11. Taylor expanded in p around 0

       \[\leadsto q + \frac{1}{2} \cdot \color{blue}{r} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.4%

       \[\leadsto \mathsf{fma}\left(0.5, r, q\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.2% accurate, 13.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 3.9 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \left(\left|r\right| + \left|p\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, r, q\_m\right)\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m)
 :precision binary64
 (if (<= q_m 3.9e-30) (* 0.5 (+ (fabs r) (fabs p))) (fma 0.5 r q_m)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (q_m <= 3.9e-30) {
		tmp = 0.5 * (fabs(r) + fabs(p));
	} else {
		tmp = fma(0.5, r, q_m);
	}
	return tmp;
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (q_m <= 3.9e-30)
		tmp = Float64(0.5 * Float64(abs(r) + abs(p)));
	else
		tmp = fma(0.5, r, q_m);
	end
	return tmp
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[q$95$m, 3.9e-30], N[(0.5 * N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * r + q$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if q < 3.9000000000000003e-30

   1. Initial program 53.3%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 12.7

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in q around 0

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

   8. Applied rewrites 16.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left|r\right| + \left|p\right|\right)} \]

   if 3.9000000000000003e-30 < q

   1. Initial program 35.5%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in q around inf

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

      1. +-commutative N/A

         \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-fabs.f64 64.9

          \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

   6. Taylor expanded in p around 0

      \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

   10. Applied rewrites 61.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r + p, 0.5, q\right)} \]

   11. Taylor expanded in p around 0

       \[\leadsto q + \frac{1}{2} \cdot \color{blue}{r} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.9%

       \[\leadsto \mathsf{fma}\left(0.5, r, q\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 13.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-43}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot r\\ \end{array} \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (if (<= r 5e-43) (* -0.5 p) (* 0.5 r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 5e-43) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
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, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    real(8) :: tmp
    if (r <= 5d-43) then
        tmp = (-0.5d0) * p
    else
        tmp = 0.5d0 * r
    end if
    code = tmp
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	double tmp;
	if (r <= 5e-43) {
		tmp = -0.5 * p;
	} else {
		tmp = 0.5 * r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	tmp = 0
	if r <= 5e-43:
		tmp = -0.5 * p
	else:
		tmp = 0.5 * r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	tmp = 0.0
	if (r <= 5e-43)
		tmp = Float64(-0.5 * p);
	else
		tmp = Float64(0.5 * r);
	end
	return tmp
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp_2 = code(p, r, q_m)
	tmp = 0.0;
	if (r <= 5e-43)
		tmp = -0.5 * p;
	else
		tmp = 0.5 * r;
	end
	tmp_2 = tmp;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := If[LessEqual[r, 5e-43], N[(-0.5 * p), $MachinePrecision], N[(0.5 * r), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.00000000000000019e-43

   1. Initial program 50.1%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in p around -inf

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

      1. lower-*.f64 5.8

         \[\leadsto \color{blue}{-0.5 \cdot p} \]

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

   if 5.00000000000000019e-43 < r

   1. Initial program 45.6%

      \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

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

      1. lower-*.f64 14.2

         \[\leadsto \color{blue}{0.5 \cdot r} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{0.5 \cdot r} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.7% accurate, 35.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \mathsf{fma}\left(0.5, r, q\_m\right) \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (fma 0.5 r q_m))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return fma(0.5, r, q_m);
}

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return fma(0.5, r, q_m)
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := N[(0.5 * r + q$95$m), $MachinePrecision]

Derivation

1. Initial program 48.5%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in q around inf

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

   1. +-commutative N/A

      \[\leadsto q \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{q \cdot \left(\frac{1}{2} \cdot \frac{\left|p\right| + \left|r\right|}{q}\right) + q \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q}} + q \cdot 1 \]

   4. *-rgt-identity N/A

      \[\leadsto \left(q \cdot \frac{1}{2}\right) \cdot \frac{\left|p\right| + \left|r\right|}{q} + \color{blue}{q} \]

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-fabs.f64 26.5

       \[\leadsto \mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \color{blue}{\left|p\right|}}{q}, q\right) \]

5. Applied rewrites 26.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(q \cdot 0.5, \frac{\left|r\right| + \left|p\right|}{q}, q\right)} \]

6. Taylor expanded in p around 0

   \[\leadsto q + \color{blue}{\frac{1}{2} \cdot \left(\left|p\right| + \left|r\right|\right)} \]
7. Step-by-step derivation

8. Applied rewrites 29.0%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left|r\right| + \left|p\right|}, q\right) \]

10. Applied rewrites 22.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(r + p, 0.5, q\right)} \]

11. Taylor expanded in p around 0

    \[\leadsto q + \frac{1}{2} \cdot \color{blue}{r} \]
12. Step-by-step derivation

13. Applied rewrites 20.9%

    \[\leadsto \mathsf{fma}\left(0.5, r, q\right) \]
14. Add Preprocessing

Alternative 10: 8.7% accurate, 41.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -0.5 \cdot p \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (* -0.5 p))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return -0.5 * p;
}

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
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, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = (-0.5d0) * p
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return -0.5 * p;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -0.5 * p

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-0.5 * p)
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -0.5 * p;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := N[(-0.5 * p), $MachinePrecision]

Derivation

1. Initial program 48.5%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in p around -inf

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

   1. lower-*.f64 5.2

      \[\leadsto \color{blue}{-0.5 \cdot p} \]

5. Applied rewrites 5.2%

   \[\leadsto \color{blue}{-0.5 \cdot p} \]
6. Add Preprocessing

Alternative 11: 1.2% accurate, 83.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ -q\_m \end{array} \]

FPCore

q_m = (fabs.f64 q)
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
(FPCore (p r q_m) :precision binary64 (- q_m))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	return -q_m;
}

Fortran

q_m =     private
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
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, r, q_m)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8), intent (in) :: r
    real(8), intent (in) :: q_m
    code = -q_m
end function

Java

q_m = Math.abs(q);
assert p < r && r < q_m;
public static double code(double p, double r, double q_m) {
	return -q_m;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	return -q_m

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return Float64(-q_m)
end

MATLAB

q_m = abs(q);
p, r, q_m = num2cell(sort([p, r, q_m])){:}
function tmp = code(p, r, q_m)
	tmp = -q_m;
end

Wolfram

q_m = N[Abs[q], $MachinePrecision]
NOTE: p, r, and q_m should be sorted in increasing order before calling this function.
code[p_, r_, q$95$m_] := (-q$95$m)

Derivation

1. Initial program 48.5%

   \[\frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q}^{2}}\right) \]
2. Add Preprocessing

3. Taylor expanded in q around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(q\right)} \]

   2. lower-neg.f64 15.2

      \[\leadsto \color{blue}{-q} \]

5. Applied rewrites 15.2%

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

Reproduce

herbie shell --seed 2024346 
(FPCore (p r q)
  :name "1/2(abs(p)+abs(r) + sqrt((p-r)^2 + 4q^2))"
  :precision binary64
  (* (/ 1.0 2.0) (+ (+ (fabs p) (fabs r)) (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q 2.0)))))))