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

Percentage Accurate: 45.7% → 70.9%

Time: 3.6s

Alternatives: 8

Speedup: 35.6×

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: 45.7% 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: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;r \leq 4.2 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(-p\right)\right)\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \mathbf{elif}\;r \leq 1.22 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + \sqrt{{\left(p - r\right)}^{2} + 4 \cdot {q\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(-p\right) + \left|r\right|\right) + \left|p\right|}{r}, 0.5, 0.5\right) \cdot r\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= r 4.2e-266)
     (* 0.5 (+ t_0 (- p)))
     (if (<= r 1.35e-89)
       (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0)))
       (if (<= r 1.22e+104)
         (*
          (/ 1.0 2.0)
          (+ t_0 (sqrt (+ (pow (- p r) 2.0) (* 4.0 (pow q_m 2.0))))))
         (* (fma (/ (+ (+ (- p) (fabs r)) (fabs p)) r) 0.5 0.5) r))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (r <= 4.2e-266) {
		tmp = 0.5 * (t_0 + -p);
	} else if (r <= 1.35e-89) {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	} else if (r <= 1.22e+104) {
		tmp = (1.0 / 2.0) * (t_0 + sqrt((pow((p - r), 2.0) + (4.0 * pow(q_m, 2.0)))));
	} else {
		tmp = fma((((-p + fabs(r)) + fabs(p)) / r), 0.5, 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)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (r <= 4.2e-266)
		tmp = Float64(0.5 * Float64(t_0 + Float64(-p)));
	elseif (r <= 1.35e-89)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	elseif (r <= 1.22e+104)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + sqrt(Float64((Float64(p - r) ^ 2.0) + Float64(4.0 * (q_m ^ 2.0))))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-p) + abs(r)) + abs(p)) / r), 0.5, 0.5) * r);
	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_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.2e-266], N[(0.5 * N[(t$95$0 + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.35e-89], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.22e+104], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[N[(p - r), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[Power[q$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-p) + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * r), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 4.19999999999999994e-266

   1. Initial program 48.3%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 65.5

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

   4. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 65.5

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

   6. Applied rewrites 65.5%

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

   if 4.19999999999999994e-266 < r < 1.34999999999999994e-89

   1. Initial program 57.1%

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

   2. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.3

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

   4. Applied rewrites 61.3%

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

   if 1.34999999999999994e-89 < r < 1.22e104

   1. Initial program 64.7%

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

   if 1.22e104 < r

   1. Initial program 21.8%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 23.2

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

   4. Applied rewrites 23.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 23.2

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 86.5%

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

       1. lift-fabs.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-+.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-fabs.f64 86.5

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

   11. Applied rewrites 86.5%

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

Alternative 2: 69.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;r \leq 4.2 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(-p\right)\right)\\ \mathbf{elif}\;r \leq 1.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \mathbf{elif}\;r \leq 9.5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \sqrt{\mathsf{fma}\left(q\_m \cdot q\_m, 4, p \cdot p\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(-p\right) + \left|r\right|\right) + \left|p\right|}{r}, 0.5, 0.5\right) \cdot r\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= r 4.2e-266)
     (* 0.5 (+ t_0 (- p)))
     (if (<= r 1.4e-89)
       (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0)))
       (if (<= r 9.5e-16)
         (* 0.5 (+ t_0 (sqrt (fma (* q_m q_m) 4.0 (* p p)))))
         (* (fma (/ (+ (+ (- p) (fabs r)) (fabs p)) r) 0.5 0.5) r))))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (r <= 4.2e-266) {
		tmp = 0.5 * (t_0 + -p);
	} else if (r <= 1.4e-89) {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	} else if (r <= 9.5e-16) {
		tmp = 0.5 * (t_0 + sqrt(fma((q_m * q_m), 4.0, (p * p))));
	} else {
		tmp = fma((((-p + fabs(r)) + fabs(p)) / r), 0.5, 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)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (r <= 4.2e-266)
		tmp = Float64(0.5 * Float64(t_0 + Float64(-p)));
	elseif (r <= 1.4e-89)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	elseif (r <= 9.5e-16)
		tmp = Float64(0.5 * Float64(t_0 + sqrt(fma(Float64(q_m * q_m), 4.0, Float64(p * p)))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-p) + abs(r)) + abs(p)) / r), 0.5, 0.5) * r);
	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_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.2e-266], N[(0.5 * N[(t$95$0 + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.4e-89], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 9.5e-16], N[(0.5 * N[(t$95$0 + N[Sqrt[N[(N[(q$95$m * q$95$m), $MachinePrecision] * 4.0 + N[(p * p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-p) + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * r), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 4.19999999999999994e-266

   1. Initial program 48.3%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 65.5

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

   4. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 65.5

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

   6. Applied rewrites 65.5%

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

   if 4.19999999999999994e-266 < r < 1.3999999999999999e-89

   1. Initial program 57.1%

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

   2. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.3

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

   4. Applied rewrites 61.3%

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

   if 1.3999999999999999e-89 < r < 9.5000000000000005e-16

   1. Initial program 63.9%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 38.3

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

   4. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 38.3

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

   6. Applied rewrites 38.3%

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

   7. Taylor expanded in r around 0

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

      1. lower-sqrt.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left({q}^{2}, 4, {p}^{2}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left(q \cdot q, 4, {p}^{2}\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left(q \cdot q, 4, {p}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \]

      7. lower-*.f64 45.2

         \[\leadsto 0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \sqrt{\mathsf{fma}\left(q \cdot q, 4, p \cdot p\right)}\right) \]

   9. Applied rewrites 45.2%

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

   if 9.5000000000000005e-16 < r

   1. Initial program 36.0%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 25.4

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

   4. Applied rewrites 25.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 25.4

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 79.1%

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

       1. lift-fabs.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-+.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-fabs.f64 79.1

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

   11. Applied rewrites 79.1%

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

Alternative 3: 70.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;r \leq 4.2 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(-p\right)\right)\\ \mathbf{elif}\;r \leq 4.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(-p\right) + \left|r\right|\right) + \left|p\right|}{r}, 0.5, 0.5\right) \cdot r\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= r 4.2e-266)
     (* 0.5 (+ t_0 (- p)))
     (if (<= r 4.3e-36)
       (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0)))
       (* (fma (/ (+ (+ (- p) (fabs r)) (fabs p)) r) 0.5 0.5) r)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (r <= 4.2e-266) {
		tmp = 0.5 * (t_0 + -p);
	} else if (r <= 4.3e-36) {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	} else {
		tmp = fma((((-p + fabs(r)) + fabs(p)) / r), 0.5, 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)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (r <= 4.2e-266)
		tmp = Float64(0.5 * Float64(t_0 + Float64(-p)));
	elseif (r <= 4.3e-36)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-p) + abs(r)) + abs(p)) / r), 0.5, 0.5) * r);
	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_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4.2e-266], N[(0.5 * N[(t$95$0 + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 4.3e-36], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-p) + N[Abs[r], $MachinePrecision]), $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 4.19999999999999994e-266

   1. Initial program 48.3%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 65.5

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

   4. Applied rewrites 65.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 65.5

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

   6. Applied rewrites 65.5%

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

   if 4.19999999999999994e-266 < r < 4.3000000000000002e-36

   1. Initial program 59.1%

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

   2. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   if 4.3000000000000002e-36 < r

   1. Initial program 37.3%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 26.2

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

   4. Applied rewrites 26.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 26.2

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

   6. Applied rewrites 26.2%

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

   7. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   9. Applied rewrites 78.1%

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

       1. lift-fabs.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-+.f64 N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-fabs.f64 78.1

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

   11. Applied rewrites 78.1%

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

Alternative 4: 65.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|p\right| + \left|r\right|\\ \mathbf{if}\;p \leq -1.35 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{2} \cdot \left(t\_0 + q\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs p) (fabs r))))
   (if (<= p -1.35e+30)
     (* 0.5 (+ t_0 (- p)))
     (if (<= p 3.3e-108) (* (/ 1.0 2.0) (+ t_0 (* q_m 2.0))) r))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(p) + fabs(r);
	double tmp;
	if (p <= -1.35e+30) {
		tmp = 0.5 * (t_0 + -p);
	} else if (p <= 3.3e-108) {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = abs(p) + abs(r)
    if (p <= (-1.35d+30)) then
        tmp = 0.5d0 * (t_0 + -p)
    else if (p <= 3.3d-108) then
        tmp = (1.0d0 / 2.0d0) * (t_0 + (q_m * 2.0d0))
    else
        tmp = 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 t_0 = Math.abs(p) + Math.abs(r);
	double tmp;
	if (p <= -1.35e+30) {
		tmp = 0.5 * (t_0 + -p);
	} else if (p <= 3.3e-108) {
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	} else {
		tmp = r;
	}
	return tmp;
}

Python

q_m = math.fabs(q)
[p, r, q_m] = sort([p, r, q_m])
def code(p, r, q_m):
	t_0 = math.fabs(p) + math.fabs(r)
	tmp = 0
	if p <= -1.35e+30:
		tmp = 0.5 * (t_0 + -p)
	elif p <= 3.3e-108:
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0))
	else:
		tmp = r
	return tmp

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	t_0 = Float64(abs(p) + abs(r))
	tmp = 0.0
	if (p <= -1.35e+30)
		tmp = Float64(0.5 * Float64(t_0 + Float64(-p)));
	elseif (p <= 3.3e-108)
		tmp = Float64(Float64(1.0 / 2.0) * Float64(t_0 + Float64(q_m * 2.0)));
	else
		tmp = 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)
	t_0 = abs(p) + abs(r);
	tmp = 0.0;
	if (p <= -1.35e+30)
		tmp = 0.5 * (t_0 + -p);
	elseif (p <= 3.3e-108)
		tmp = (1.0 / 2.0) * (t_0 + (q_m * 2.0));
	else
		tmp = 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_] := Block[{t$95$0 = N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -1.35e+30], N[(0.5 * N[(t$95$0 + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 3.3e-108], N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$0 + N[(q$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], r]]]

Derivation

1. Split input into 3 regimes

2. if p < -1.3499999999999999e30

   1. Initial program 33.8%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.3

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

   6. Applied rewrites 71.3%

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

   if -1.3499999999999999e30 < p < 3.3000000000000002e-108

   1. Initial program 58.6%

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

   2. Taylor expanded in q around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.7

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

   4. Applied rewrites 56.7%

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

   if 3.3000000000000002e-108 < p

   1. Initial program 34.4%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 78.7%

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

Alternative 5: 64.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;p \leq -8.5 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(-p\right)\right)\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{fma}\left(r + p, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 (<= p -8.5e+24)
   (* 0.5 (+ (+ (fabs p) (fabs r)) (- p)))
   (if (<= p 3.3e-108) (fma (+ r p) 0.5 q_m) r)))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double tmp;
	if (p <= -8.5e+24) {
		tmp = 0.5 * ((fabs(p) + fabs(r)) + -p);
	} else if (p <= 3.3e-108) {
		tmp = fma((r + p), 0.5, q_m);
	} else {
		tmp = 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 (p <= -8.5e+24)
		tmp = Float64(0.5 * Float64(Float64(abs(p) + abs(r)) + Float64(-p)));
	elseif (p <= 3.3e-108)
		tmp = fma(Float64(r + p), 0.5, q_m);
	else
		tmp = r;
	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[p, -8.5e+24], N[(0.5 * N[(N[(N[Abs[p], $MachinePrecision] + N[Abs[r], $MachinePrecision]), $MachinePrecision] + (-p)), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 3.3e-108], N[(N[(r + p), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision], r]]

Derivation

1. Split input into 3 regimes

2. if p < -8.49999999999999959e24

   1. Initial program 34.3%

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

   2. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\left|p\right| + \left|r\right|\right) + \left(\mathsf{neg}\left(p\right)\right)\right) \]

      2. lower-neg.f64 70.9

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

   4. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.9

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

   6. Applied rewrites 70.9%

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

   if -8.49999999999999959e24 < p < 3.3000000000000002e-108

   1. Initial program 58.4%

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

   2. Taylor expanded in q around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in q around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-+.f64 54.8

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

   7. Applied rewrites 54.8%

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

   if 3.3000000000000002e-108 < p

   1. Initial program 34.4%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 16.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 16.0%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 78.7%

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

Alternative 6: 54.5% accurate, 35.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 4.5 \cdot 10^{-10}:\\ \;\;\;\;q\_m\\ \mathbf{else}:\\ \;\;\;\;r\\ \end{array} \end{array} \]

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 4.5e-10) q_m 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 <= 4.5e-10) {
		tmp = q_m;
	} else {
		tmp = 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 <= 4.5d-10) then
        tmp = q_m
    else
        tmp = 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 <= 4.5e-10) {
		tmp = q_m;
	} else {
		tmp = 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 <= 4.5e-10:
		tmp = q_m
	else:
		tmp = 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 <= 4.5e-10)
		tmp = q_m;
	else
		tmp = 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 <= 4.5e-10)
		tmp = q_m;
	else
		tmp = 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, 4.5e-10], q$95$m, r]

Derivation

1. Split input into 2 regimes

2. if r < 4.5e-10

   1. Initial program 53.6%

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

   2. Taylor expanded in q around inf

      \[\leadsto \color{blue}{q} \]
   3. Step-by-step derivation

   4. Applied rewrites 44.1%

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

   if 4.5e-10 < r

   1. Initial program 35.7%

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

   2. Taylor expanded in p around inf

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

   4. Applied rewrites 14.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 13.8%

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

   7. Taylor expanded in p around -inf

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

   9. Applied rewrites 67.9%

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

Alternative 7: 34.6% accurate, 250.0× speedup

Math

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

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 r)

C

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

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 = r
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 r;
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in p around inf

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

4. Applied rewrites 8.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in p around -inf

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

9. Applied rewrites 34.6%

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

Alternative 8: 1.8% accurate, 250.0× speedup

Math

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

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 p)

C

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

Python

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

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return 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 = 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_] := p

Derivation

1. Initial program 45.7%

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

2. Taylor expanded in p around inf

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

4. Applied rewrites 8.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 7.5%

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

7. Taylor expanded in p around inf

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

9. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025105 
(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)))))))