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

Percentage Accurate: 44.0% → 82.3%

Time: 3.5s

Alternatives: 9

Speedup: 5.1×

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: 44.0% 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: 82.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, \left(\left(-p\right) + r\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= q_m 1.3e+118) (fma t_0 0.5 (* (+ (- p) r) 0.5)) (fma t_0 0.5 q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 1.3e+118) {
		tmp = fma(t_0, 0.5, ((-p + r) * 0.5));
	} else {
		tmp = fma(t_0, 0.5, q_m);
	}
	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(r) + abs(p))
	tmp = 0.0
	if (q_m <= 1.3e+118)
		tmp = fma(t_0, 0.5, Float64(Float64(Float64(-p) + r) * 0.5));
	else
		tmp = fma(t_0, 0.5, 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_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 1.3e+118], N[(t$95$0 * 0.5 + N[(N[((-p) + r), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 1.30000000000000008e118

   1. Initial program 56.6%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 72.3

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 84.7

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

   7. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 84.8%

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

   if 1.30000000000000008e118 < q

   1. Initial program 16.5%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 24.1

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 30.1

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

   7. Applied rewrites 30.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 30.2%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 77.0%

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

Alternative 2: 82.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;\left(t\_0 + \left(r - p\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= q_m 1.3e+118) (* (+ t_0 (- r p)) 0.5) (fma t_0 0.5 q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 1.3e+118) {
		tmp = (t_0 + (r - p)) * 0.5;
	} else {
		tmp = fma(t_0, 0.5, q_m);
	}
	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(r) + abs(p))
	tmp = 0.0
	if (q_m <= 1.3e+118)
		tmp = Float64(Float64(t_0 + Float64(r - p)) * 0.5);
	else
		tmp = fma(t_0, 0.5, 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_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 1.3e+118], N[(N[(t$95$0 + N[(r - p), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 1.30000000000000008e118

   1. Initial program 56.6%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 72.3

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

   4. Applied rewrites 72.3%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 84.7

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

   7. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 84.7

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

   9. Applied rewrites 84.7%

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

   10. Taylor expanded in r around 0

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

       1. lower--.f64 84.7

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

   12. Applied rewrites 84.7%

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

   if 1.30000000000000008e118 < q

   1. Initial program 16.5%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 24.1

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 30.1

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

   7. Applied rewrites 30.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 30.2%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 77.0%

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

Alternative 3: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;p \leq -4.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, -0.5 \cdot p\right)\\ \mathbf{elif}\;p \leq -8.4 \cdot 10^{-256}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + r\right) \cdot 0.5\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= p -4.8e+28)
     (fma t_0 0.5 (* -0.5 p))
     (if (<= p -8.4e-256) (fma t_0 0.5 q_m) (* (+ t_0 r) 0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (p <= -4.8e+28) {
		tmp = fma(t_0, 0.5, (-0.5 * p));
	} else if (p <= -8.4e-256) {
		tmp = fma(t_0, 0.5, q_m);
	} else {
		tmp = (t_0 + r) * 0.5;
	}
	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(r) + abs(p))
	tmp = 0.0
	if (p <= -4.8e+28)
		tmp = fma(t_0, 0.5, Float64(-0.5 * p));
	elseif (p <= -8.4e-256)
		tmp = fma(t_0, 0.5, q_m);
	else
		tmp = Float64(Float64(t_0 + r) * 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_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -4.8e+28], N[(t$95$0 * 0.5 + N[(-0.5 * p), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, -8.4e-256], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision], N[(N[(t$95$0 + r), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -4.79999999999999962e28

   1. Initial program 31.1%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 81.1

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

   7. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in p around -inf

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

       1. lift-*.f64 72.1

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

   12. Applied rewrites 72.1%

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

   if -4.79999999999999962e28 < p < -8.40000000000000011e-256

   1. Initial program 59.5%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 51.4

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

   7. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 51.4%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 59.3%

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

   if -8.40000000000000011e-256 < p

   1. Initial program 46.6%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 64.1

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 64.1

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

   9. Applied rewrites 64.1%

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

   10. Taylor expanded in p around 0

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

   12. Applied rewrites 63.2%

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

Alternative 4: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;p \leq -4.8 \cdot 10^{+28}:\\ \;\;\;\;\left(t\_0 + \left(-p\right)\right) \cdot 0.5\\ \mathbf{elif}\;p \leq -8.4 \cdot 10^{-256}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + r\right) \cdot 0.5\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= p -4.8e+28)
     (* (+ t_0 (- p)) 0.5)
     (if (<= p -8.4e-256) (fma t_0 0.5 q_m) (* (+ t_0 r) 0.5)))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (p <= -4.8e+28) {
		tmp = (t_0 + -p) * 0.5;
	} else if (p <= -8.4e-256) {
		tmp = fma(t_0, 0.5, q_m);
	} else {
		tmp = (t_0 + r) * 0.5;
	}
	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(r) + abs(p))
	tmp = 0.0
	if (p <= -4.8e+28)
		tmp = Float64(Float64(t_0 + Float64(-p)) * 0.5);
	elseif (p <= -8.4e-256)
		tmp = fma(t_0, 0.5, q_m);
	else
		tmp = Float64(Float64(t_0 + r) * 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_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, -4.8e+28], N[(N[(t$95$0 + (-p)), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[p, -8.4e-256], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision], N[(N[(t$95$0 + r), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -4.79999999999999962e28

   1. Initial program 31.1%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 81.1

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

   7. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 81.1

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

   9. Applied rewrites 81.1%

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

   10. Taylor expanded in p around -inf

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 72.1

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

   12. Applied rewrites 72.1%

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

   if -4.79999999999999962e28 < p < -8.40000000000000011e-256

   1. Initial program 59.5%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 51.4

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

   7. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 51.4%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 59.3%

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

   if -8.40000000000000011e-256 < p

   1. Initial program 46.6%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 64.1

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 64.1

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

   9. Applied rewrites 64.1%

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

   10. Taylor expanded in p around 0

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

   12. Applied rewrites 63.2%

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

Alternative 5: 59.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} t_0 := \left|r\right| + \left|p\right|\\ \mathbf{if}\;q\_m \leq 1.1 \cdot 10^{+72}:\\ \;\;\;\;\left(t\_0 + r\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 0.5, q\_m\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (+ (fabs r) (fabs p))))
   (if (<= q_m 1.1e+72) (* (+ t_0 r) 0.5) (fma t_0 0.5 q_m))))

C

q_m = fabs(q);
assert(p < r && r < q_m);
double code(double p, double r, double q_m) {
	double t_0 = fabs(r) + fabs(p);
	double tmp;
	if (q_m <= 1.1e+72) {
		tmp = (t_0 + r) * 0.5;
	} else {
		tmp = fma(t_0, 0.5, q_m);
	}
	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(r) + abs(p))
	tmp = 0.0
	if (q_m <= 1.1e+72)
		tmp = Float64(Float64(t_0 + r) * 0.5);
	else
		tmp = fma(t_0, 0.5, 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_] := Block[{t$95$0 = N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[q$95$m, 1.1e+72], N[(N[(t$95$0 + r), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * 0.5 + q$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if q < 1.1e72

   1. Initial program 56.8%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 87.3

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

   7. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lift-fabs.f64 N/A

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

      11. lift-fabs.f64 N/A

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

      12. metadata-eval 87.3

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

   9. Applied rewrites 87.3%

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

   10. Taylor expanded in p around 0

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

   12. Applied rewrites 50.8%

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

   if 1.1e72 < q

   1. Initial program 23.7%

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

   2. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in p around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 36.1

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

   7. Applied rewrites 36.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-in N/A

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

      8. lower-fma.f64 N/A

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

   9. Applied rewrites 36.2%

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

   10. Taylor expanded in q around inf

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

   12. Applied rewrites 72.0%

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

Alternative 6: 45.7% accurate, 5.1× speedup

Math

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

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 (+ (fabs r) (fabs p)) 0.5 q_m))

C

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

Julia

q_m = abs(q)
p, r, q_m = sort([p, r, q_m])
function code(p, r, q_m)
	return fma(Float64(abs(r) + abs(p)), 0.5, 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[(N[(N[Abs[r], $MachinePrecision] + N[Abs[p], $MachinePrecision]), $MachinePrecision] * 0.5 + q$95$m), $MachinePrecision]

Derivation

1. Initial program 44.0%

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

2. Taylor expanded in r around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. mul-1-neg N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 57.1

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in p around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 67.5

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

7. Applied rewrites 67.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

9. Applied rewrites 67.6%

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

10. Taylor expanded in q around inf

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

12. Applied rewrites 45.7%

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

Alternative 7: 37.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 7.4 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot r\\ \mathbf{else}:\\ \;\;\;\;q\_m\\ \end{array} \end{array} \]

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 7.4e-97) (* 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 <= 7.4e-97) {
		tmp = 0.5 * r;
	} else {
		tmp = q_m;
	}
	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 (q_m <= 7.4d-97) then
        tmp = 0.5d0 * r
    else
        tmp = q_m
    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 (q_m <= 7.4e-97) {
		tmp = 0.5 * r;
	} else {
		tmp = q_m;
	}
	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 q_m <= 7.4e-97:
		tmp = 0.5 * r
	else:
		tmp = 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 <= 7.4e-97)
		tmp = Float64(0.5 * r);
	else
		tmp = q_m;
	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 (q_m <= 7.4e-97)
		tmp = 0.5 * r;
	else
		tmp = q_m;
	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[q$95$m, 7.4e-97], N[(0.5 * r), $MachinePrecision], q$95$m]

Derivation

1. Split input into 2 regimes

2. if q < 7.39999999999999951e-97

   1. Initial program 55.0%

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

   2. Taylor expanded in r around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{r} \]

      3. metadata-eval 10.6

         \[\leadsto 0.5 \cdot r \]

   4. Applied rewrites 10.6%

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

   if 7.39999999999999951e-97 < q

   1. Initial program 38.4%

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

   2. Taylor expanded in q around inf

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

   4. Applied rewrites 49.9%

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

Alternative 8: 36.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} q_m = \left|q\right| \\ [p, r, q_m] = \mathsf{sort}([p, r, q_m])\\ \\ \begin{array}{l} \mathbf{if}\;q\_m \leq 7.2 \cdot 10^{-134}:\\ \;\;\;\;-0.5 \cdot p\\ \mathbf{else}:\\ \;\;\;\;q\_m\\ \end{array} \end{array} \]

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 7.2e-134) (* -0.5 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 <= 7.2e-134) {
		tmp = -0.5 * p;
	} else {
		tmp = q_m;
	}
	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 (q_m <= 7.2d-134) then
        tmp = (-0.5d0) * p
    else
        tmp = q_m
    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 (q_m <= 7.2e-134) {
		tmp = -0.5 * p;
	} else {
		tmp = q_m;
	}
	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 q_m <= 7.2e-134:
		tmp = -0.5 * p
	else:
		tmp = 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 <= 7.2e-134)
		tmp = Float64(-0.5 * p);
	else
		tmp = q_m;
	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 (q_m <= 7.2e-134)
		tmp = -0.5 * p;
	else
		tmp = q_m;
	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[q$95$m, 7.2e-134], N[(-0.5 * p), $MachinePrecision], q$95$m]

Derivation

1. Split input into 2 regimes

2. if q < 7.1999999999999998e-134

   1. Initial program 54.0%

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

   2. Taylor expanded in p around -inf

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

      1. lower-*.f64 11.0

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

   4. Applied rewrites 11.0%

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

   if 7.1999999999999998e-134 < q

   1. Initial program 40.2%

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

   2. Taylor expanded in q around inf

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

   4. Applied rewrites 47.0%

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

Alternative 9: 35.8% accurate, 56.9× 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 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 44.0%

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

2. Taylor expanded in q around inf

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

4. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2025116 
(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)))))))